> Loss or profit only happens when you sell your assets.
This is a meme phrase; losses absolutely do happen before you sell your assets. (If I bought Enron at $90 and never sold, I didn't lose any money, right?)
> "American Airlines spent $13 billion on share buybacks for 10 years through 2019". They should have used that money more wisely. Why should the taxpayers give money away
You have two misunderstandings here. The Fed is not bailing out American Airlines by giving money away. That has to go through Congress. (I have no claims about whether bailing out AA is a good decision or not). Bailouts also generally are not free giveaways; the government often takes equity stakes.
> or lent money at close to zero interest
The Fed is lending everyone money at 0% to avoid a recession. Regardless of whether AA previously bought back their stock, they could still borrow at 0%. The Fed is doing this because the alternative (letting everything crash) is much worse.
> when the companies had the money and dedicated it to artificially increase share prices?
Share buybacks do not "manipulate the market" (not your words, but common phrasing among anti-buyback advocates) and artificially increase share prices. They return capital to shareholders. The share price goes up because the stock now represents a better return on investment (when AA buys back stock, it no longer a bunch of market cap tied up in money uselessly sitting around in a money market account).
When times get tough, AA can sell equity for more money, which is a stock buyback but in reverse. That's what could happen here. This ebb and flow (raising capital by issuing shares when you need it, returning capital when you don't) is the whole point of equity markets.
See https://www.bloomberg.com/opinion/articles/2020-03-17/the-go... for a more comprehensive argument about why share buybacks for airlines are more rational than holding money for a rainy day.