Is this method of getting free money viable and consistent with law?

There is a loophole within a loophole.
But first - 24 months is way too much. With CC you're taking short time credit. And short-time in this case means days, very rarely few months. Never year or two.

But even with, let say, 52 days you could think about doing some low-interest deals. You have your billing period, 1st to 30 day of the month, PLUS debt repayment period, additional 22 days. So, in theory nothing stops you from buying 30 days bonds with CC.

The problem is that bonds (or any other interest thing) are calculated exactly as your billing period WITH the catch that the money must be IN at the begining of first day. SO you would need to buy them at the last day of PREVIOUS month and stick till first day of the next month to get dyvidend. Let say investment day 31st of May, hold bond for whole June, get paid 1st of July.
But you need to pay back the CC 21st of June the latest.

Now, doing that "hoovering" is called Credit card kiting and of course require very strict planning. With two cards you can see you could pay that one debt with another CC prolonging pay off day with extra 31 days (remaining 9 days of June and 22 of July). What's stopping you? Usually you cannot pay of CC with a CC directly.
As you imagined you would need to establish some place where you could pay and get the money. Now, any place that allow CC payment will tell you they have to pay commission for any payment done. So as the owner of the terminal you pay the commission, you need to pay the tax on that art you sold and so on. And that might (and with 100% certainty is) higher that what you would earn on some short time investment (like 3 months for example to do some real kiting).

Is this method of getting free money viable and consistent with law?

Yes, it's usually withing the bank right to spot such things but they usually don't as people who try kiting slip very often and then give bank hefty gain.

If so, why does the financial system have this silly loophole?

Because those who would earn money that way earn very little compared to all the ones who tries and fails and need to pay the bank. It's a gamble, banks like to play those when they are in the position of power (if they spot more people gaining on this they can just change rules).

If not, what kinds of laws or catch-22's prevent this kind of thing?

Nothing, banks looks at such idea and say "Try, prove to us that you can earn more that what you would need pay to us".

I have no idea if you can do this. But I would like to point out that to what I see as missing parts of your proposed plan.

1) There is a cost to setting up the company, and then getting it into a place to accept credit cards.

2) You covered the cost that the purchaser (you) has to pay for the transfer. But you didn't cover the cost of the initial transaction for you the seller. Also do you know if the merchant account has a maximum amount before they want more information about your business?

3) Taxes. Your business purchased that art for almost nothing, and then sold if for a large profit. The national and maybe even the local government wants their taxes.

4) The monthly payment. The credit card will expect you to make payments even when the interest rate is zero.

5) You want to do this on a card that is not used for anything else. You don't want anything to happen that will trigger all you other transactions to incur interest charges.

It can be done, sort of, or could be done a decade ago*, and without the business aspect. I know because I did it, and it was fairly commonly known among frugalistas like the Money Moustache community.

The "sort of" refers to the borrowing money on the credit card. I don't know of any card that would let you get a cash advance at 0%. Instead, what you do is get a new card with a 0% introductory period (and often a sign-up bonus), put all your chargable spending on that card until the end of the 0% period, then either pay it off or transfer it to another card. Back then, you could even find cards with no-cost balance transfers. Then repeat the cycle with another card.

Meanwhile, your money stays invested in the market, and remember that this was in the couple of years after the '08 market crash, when I was seeing 15-20% or more annual return on my mutual funds. (Given current market conditions, I wouldn't try it today, since I'd expect little or no long-term growth.)

*I don't think it's really viable currently, as CC issuers have tightened things up quite a bit.

I did this twice many years ago, it was completely legal and it worked for me. I wouldn't do again.

Here is what I did:

• The offer was 0% on balance transfers (no fees) for 2 years
• I applied for a transfer and used my debit card number
• I was approved and money was deposited in my checking account
• I calculated the total minimum payments for the 2 years and put that in my savings account
• I bought a CD with a 2 year term with the remainder
• I made no purchases on the new credit card
• I made minimum payments each month from my savings account
• After 6 months of on time payments, I was allowed to move my payment day forward
• When the CD matured, I paid the remaining balance and closed the line of credit

This made sense because the CDs were FDIC insured with a fixed interest rate. If I hadn't been able to move the billing date forward, I would have lost 3 months interest on the CDs. I don't remember exactly how much I made, but maybe around $700 for the 2 cards combined.

It was fun to do and I was kind of proud of myself for making it work, but I wouldn't do it again now for a few reasons

1. Interest rates are low. My bank will only give me 0.65% on a 2 year CD presently, so I would have to invest in something that isn't guaranteed for a meaningful return.
2. Creditors may be wise to the debit card transfer that equated to a cash advance in my case and may limit a balance transfer amount to your current balance.
3. Minimum payments are higher as rule now, so that would leave less money to invest.
4. If I tried to do this enough to make any real money, I would damage my credit and the interest free offers would dry up.
5. If I need some extra cash, I can just get a part-time job for 2 months instead of waiting 2 years.