Why is the bid-ask spread considered a cost?

This is a misconception.

One of the explanations is that if you buy at the ask price and want to sell it right away, you can only sell at the bid price.

This is incorrect. There are no two separate bid and ask prices. The price you buy (your "bid") is the same price someone else sells (their "sell"). The same goes when you sell - the price you sell at is the price someone else buys.

There's no spread with stocks. Emphasized it on purpose, because many people (especially those who gamble on stock exchange without knowing what they're doing) don't understand how the stock market works. On the stock exchange, the transaction price is the match between the bid price and the ask price. Thus, on any given transaction, bid always equals ask. There's no spread.

There is spread with commodities (if you buy it directly, especially), contracts, mutual funds and other kinds of brokered transactions that go through a third party. The difference (spread) is that third party's fee for assuming part of the risk in the transaction, and is indeed added to your cost (indirectly, in the way you described). These transactions don't go directly between a seller and a buyer.

For example, there's no buyer when you redeem some of your mutual fund - the fund pays you money. So the fund assumes certain risk, which is why there's a spread in the prices to invest and to redeem. Similarly with commodities: when you buy a gold bar - you buy it from a dealer, who needs to keep a stock. Thus, the dealer will not buy from you at the same price: there's a premium on sale and a discount on buy, which is a spread, to compensate the dealer for the risk of keeping a stock.

As an aside, on most securities with a spread of the minimum tick, there would be no bid ask spread if so-called "locked markets", where the price of the best bid on one exchange is equal to the price of the best ask on another, were permitted.

It is currently forbidden for a security to have posted orders having the same price for both bid and ask even though they're on different exchanges.

Option spreads would narrow as well as a result.

There's a lively example that might be able to prove it to some extent.

Airport shows the current exchange rate info, EUR 1 = USD 1.3/USD 1.4, 1.4 represents limited ask price online, 1.3 represents limited bid price online.

Assumes there are only two tourists.

Kate, who wants to go to Europe, needs 500 euro. She has to pay 500*1.4=700 US dollars.

At the same time, John, who wants to go back to USA, needs to exchange the remained 500 euro into US dollars. He will get 500*1.3=650 US dollars.

The 50 dollars' difference is generated at the moment. It is the revenue of airport dealer who have the deal done, paid by the pair tourists. We can also call it transaction cost on Kate's or John's side. And the average transaction cost is 25 US dollars on each side. Strictly speaking, it is a relative 50 US dollars' transaction cost against peer-to-peer or face-to-face transaction between buyer and seller.

Generally speaking, the transaction cost is untaken by all buyers/sellers who participated the dealing within the market, earned by the dealer at the same time.

Hopes it helps in some way~

The statement that There are no two separate bid and ask prices is nonsensical.

Try this exercise. XYZ has a bid/ask of $49.75 x $50.00. Execute two market orders at current price before the quote changes. You'll buy 100 shares at the ask price of $ $50.00 and you'll sell sell 100 shares at the bid price of $49.75. Because you have executed offsetting positions, there will be no new equity position in your account but there will be $25 less in it. That spread is the vig that the market maker collected from you for trading at the market.

In this example, if you just bought 100 shares at $50.00, then it would require the stock to rise 25 cents in order for you to break even and sell at the new/higher bid price of $50.00 (ignoring commissions). While some may prefer to play word games and claim that this isn't a cost, it is indeed a cost because you have paid more for the security than what it's worth on the open market and that amount is the spread.