I challenge the question. You see the minimum price rule mentioned a lot. But it is only one rule in a set of interlocking rules. You can't point to one of of a long list of requirements and say why is rule X set to Y value, while ignoring the rest.

for example the New York Stock Exchange (NYSE) has these requirements:

Required to meet all of the following public distribution criteria:

• Shareholders 300
• Shares 200,000
• Market value $1MM

FINANCIAL CRITERIA Required to meet all of the following*:

• I. $2MM in stockholders’ equity if reported losses from continuing operations and/or net losses in 2 of last 3 fiscal years
• II. $4MM stockholders’ equity if reported losses from continuing operations and/or net losses in 3 of last4 fiscal years
• III. $6MM in stockholder’s equity if reported losses from continuing operations and/or net losses in last 5fiscal years
• IV. Financial condition of the company cannot be impaired

*The NYSE MKT will not normally suspend an issuer which is below criteria (I) through (III)above if issuer is in compliance with all of the following:

• 1.1MM publicly-held shares
• $15MM market value of publicly-held shares
• 400 round lot holders
• $50MM market cap OR total assets & revenue of $50MM each in last fiscal year or 2 of last 3 fiscal years

On the quoted source a specific minimum list price isn't specified but the following is mentioned in the section:

NYSE MKT reserves the right to delist companies should any of the following occur: Common stock sells for a low price for a substantial period of time and/or issuer fails to effect a reverse split of such shares within a reasonable time after being notified of such potential actions by the Exchange

from your question:

From my observations, some reverse stock splits happen just because companies want to raise their share price to comply with the stock exchange's minimum share price regulations.

The NYSE wants them to do reverse stock split to stay in the market. They allow this. They encourage it, as long as they also meet the other standards.

They care about the market value, the number of shares, the number of shareholders, and are they profitable. It isn't important that the number is $1 or and other value. The important thing is that they do have a several metrics and rules.

One possible reason is to allow reasonable granularity of price movements. With a price of 20 cents, a one cent change in price would be a 5% change, so requiring higher prices would allow for more granularity and less variance. Sure, computers can deal with more than two decimals of precision, but humans that are used to units and hundredths in most currencies aren't so adaptable.

Another possible reason (that's also overcomable) is the effect of transaction costs. With lower prices, lots of 100 shares can be purchased for small amounts, and transactions costs would be a more significant part.

In reality, the biggest factors are likely inertia, legacy, and tradition - certainly computers make some of these obstacles easier to overcome, but markets like this that are a hundred years old can be slower to adopt change without having some sort of value added by the change.

TL,DR: A minimum share price is part of the criteria exchanges use to weed-out failing companies. Falling below a minimum stock-price threshold does not in itself mean the company has, or will, fail. It is, however, an indicator of potential trouble that needs to be addressed.

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Exchanges generally only want to list (reasonably) successful companies. As Investopedia says (emphasis mine):

Understanding Listing Requirements

Listing requirements are a set of conditions which a firm must meet before listing a security on one of the organized stock exchanges, such as the New York Stock Exchange (NYSE), the Nasdaq, the London Stock Exchange, or the Tokyo Stock Exchange. The requirements typically measure the size and market share of the security to be listed, and the underlying financial viability of the issuing firm. Exchanges establish these standards as a means of maintaining their own reputation and visibility.

Investopedia: Listing Requirements

Major stock exchanges, like the Nasdaq, are exclusive clubs—their reputations rest on the companies they trade. As such, the Nasdaq won't allow just any company to be traded on its exchange. Only companies with a solid history and top-notch management behind them are considered.

Investopedia: What are the listing requirements for NASDAQ?

(Whether this stance is to "protect the investor" by rejecting financially-risky companies, or to maintain the reputation of the exchange is probably a matter of debate: in reality it is probably a bit of both).

While the minimum price-per-share is by no means the only criterion for gauging a company's financial health, it is one of the measures that can be used to try and weed-out failing or poorly-performing companies. If a company was originally listed on, say, the NYSE at a price healthily above $4, but the price has dropped to around $1 for an extended period, then usually this indicates something is going wrong with the financial health of the company. Whether this is caused by bad luck, bad management, or by other reasons doesn't really matter: unless something is done to reverse the trend there is a danger that the price will drop further and become "penny shares".

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Reverse Stock Splits are not a "Quick Fix"

You say in the question:

From my observations, some reverse stock splits happen just because companies want to raise their share price to comply with the stock exchange's minimum share price regulations. Given that the market capitalization, value, and operations of companies aren't affected by stock splits, the minimum price requirement just imposes additional paperwork and administrative expenses on companies.

While the market capitalization is unaffected by a reverse stock split, the same cannot be said for confidence in the company. From Investopedia:

Market Impact of Reverse Stock Splits

Generally, a reverse stock split is not perceived positively by market participants. It indicates that the stock price has gone to the bottom and the company management is attempting to inflate the prices artificially without any real business proposition.

And from Wikipedia:

There is a stigma attached to doing a reverse stock split, as it underscores the fact that shares have declined in value, so it is not common and may take a shareholder or board meeting for consent.

If the alternative is de-listing, many companies may decide to perform a reverse stock split, despite the negative connotations. However, while this may bring the share price back in line with an exchange's rules, just doing a reverse split will not make a company more successful.

A reverse stock split may ease the symptom (low share price) but will not cure an underlying disease.

If no other action is taken it may just be "postponing the inevitable". On the other hand, it could give the company time to "ride out a rough patch" (if simply a victim of adverse market conditions), or be the necessary "kick in the pants" to put in place measures to turn things around (restructuring the company, cutting costs, finding new markets etc.). However, simply doing a reverse split alone will generally not do the trick.

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Evidence in Support of "Successful Companies Only"

It is hard to find a quote that explicitly states what the intention of a stock exchange's rules are (let alone the specific rule on minimum share price). Because such rules form a contract between the companies being listed and the exchanges, they are written in "legalese", where ambiguity is anathema. A "dry" list of rules will, ideally, provide an objective set of criteria that a given company meets or does not meet. Trying to explain the reasoning behind the rules is a much more subjective exercise which can often be open to interpretation and is generally avoided in legal documents. For instance, Section 8 Suspension and Delisting of the NYSE's Listed Company Manual ("the Exchange's basic handbook of policies, practices and procedures for listed companies") sets out all the applicable rules, but says very little about why those rules are there (several quotes will be taken from this document).

In the absence of an explicitly-stated reason, there is going to be a certain amount of "reading between the lines" and inferring the subjective "why" from the objective "what".

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The "policy" section of the above-mentioned NYSE document opens with:

801.00 Policy

Securities admitted to the list may be suspended from dealings or removed from the list at any time that a company falls below certain quantitative and qualitative continued listing criteria. When a company falls below any criterion, the Exchange will review the appropriateness of continued listing.

While one might debate precisely how indicative meeting each criteria is of "a successful company", the "quantitative" criteria in the document (e.g. minimum share price, minimum number of stockholders, minimum share capital etc.) tend to be tests that are passed by a successful company, and more likely to be failed by a company in trouble.

Other criteria that may trigger (or initiate the process leading to) a de-listing include (all from section 802.01D Other Criteria of the above NYSE document):

Reduction in Operating Assets and/or Scope of Operations

The operating assets have been or are to be substantially reduced such as by sale, lease, spin off, distribution, discontinuance, abandonment, destruction, condemnation, seizure or expropriation, or the company has ceased to be an operating company or discontinued a substantial portion of its operations or business for any reason whatsoever [...]

Bankruptcy and/or Liquidation

An intent to file under any of the sections of the bankruptcy law has been announced or a filing has been made or liquidation has been authorized and the company is committed to proceed.

Authoritative Advice Received that Security is Without Value

Advice has been received, deemed by the Exchange to be authoritative, that the security is without value.

Delisting will be considered when [selected entries]:

• The failure of a company to make timely, adequate, and accurate disclosures of information to its shareholders and the investing public.
• Failure to observe good accounting practices in reporting of earnings and financial position.
• Other conduct not in keeping with sound public policy.
• Unsatisfactory financial conditions and/or operating results.
• Inability to meet current debt obligations or to adequately finance operations.
• Abnormally low selling price or volume of trading.

and the somewhat "catch-all":

• Any other event or condition which may exist or occur that makes further dealings or listing of the securities on the Exchange inadvisable or unwarranted in the opinion of the Exchange.

To my mind, all these criteria and rules are clearly about identifying (and then "weeding-out") companies that are in poor financial health. One of these criteria is concerned with the minimum share price: it seems entirely justified to assert that the reason for a minimum share price is to (help) identify poorly performing companies.

To keep the vig charged for trades small enough so they can happen.

Imagine you own a grocery store and a supplier puts bottled water on your shelves. To offer some variety, they have bottles that contain 1l of water and other bottles that hold only 1ml of water.

Customers would still want to buy 1l of water, so some buy the 1l bottle while others buy the 1ml bottle.

As the store owner you would quickly realize that the 1ml bottles are actually a huge loss for you, because each bottle has to be handled, restocked, scanned at the checkout, etc. So you define a minimum bottle size (or minimum sales commission, which is effectively the same) as requirement for supplies to put onto your shelves.

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This is of course over-simplified, any may represent just one of many aspects for minimum share price requirements, but it surely is one aspect because smaller stocks mean more computational transactions. Computing costs are in fact a huge price factor for stock exchanges.

There is no rational reason for this at all. It is just a matter of tradition.

Compare the stocks in the US NASDAQ 100 index with the UK FTSE 100 index, for example.

Half the stocks in the FTSE 100 have prices less than £10, and only one has a price just over £100. There are a few major international companies in the FTSE with prices less than £2 and nobody is suggesting they ought to do reverse stock splits to change that. In fact, it is much more common in the UK to do "normal" stock splits to keep share prices low, not high.

On the other hand, half the stocks in the NASDAQ 100 have prices over $100, and a few are over $1,000.

If you want to argue that somehow the NASDAQ is a "better" exchange than the London stock exchange because stock prices are so much higher, feel free to display your national prejudices!

Penny stocks are prone to pump and dump scams

In addition to the excellent answers here, I would like to add this somewhat dark stock exchange experience.

An enterprise that falls from grace will have a falling price, falling liquidity and no coverage. These factors may work in a vicious cycle until the stock is trading for cents with almost no liquidity.

A stock with low price and light liquidity is prime material for a pump and dump scheme.

So a stock exchange then has two choices:

1. Preserve and harbor stock issues that attract criminal activity; or

2. Weed out stocks with these characteristics;

It's not a hard choice.