Bailouts go to banks because the government is on the hook for insuring cash deposits anyway. Do you have any examples of hedge funds being bailed out?
Now, granted, public funds weren't used for the bailout, in this case it was more the Fed organizing private banks for the bailout. Also, the investors in LTCM still lost a ton of money.
But LTCM is still a great prototypical example of "vacuuming up nickels in front of a steamroller", i.e. these traders can take these positions (LTCM was focused on merger arbitrage IIRC) that seem like a license to print money, except they invariably can lead to some giant risk blowup. And when they fail, they have a knack for taking down the whole financial system with them so the government has to get involved (even if it is just cajoling other bankers in a room) to bail them out.
They were able to exploit their prime brokers to give them tons of leverage at essentially zero costs. Since LTCM was one of the first, the banks didn't really know how to model their counterparty risk and they were dazzled by the prestige. Since that occurred there's been a ton more hedge funds, and banks have way better systems for managing that level of counterparty exposure. Plus the post-GFC regulations severely limit bank leverage anyway, so they couldn't give cheap leverage to their hedge fund clients even if they wanted to.
Again that's why we've never seen another LTCM in nearly a quarter of a century. Pointing to LTCM as an example of risk in the financial system of 2021 would be like pointing to Chernobyl as an example of risks in modern nuclear reactors.
cough Lehman Brothers went down in 2008.
(tho was more about buying the good assets to avoid adding even more fire to the crisis, I think investors were mostly wiped out)