That's what you might think, but since the Volcker rule, a lot of the kind of 'liquidity provision' (aka trading) done in the banks has now been forced into funds like Citadel.

Meaning 'too big to fail' applies to hedge funds now just like investment banks. Ala LTCM

https://en.wikipedia.org/wiki/When_Genius_Failed

Goldman received $10B in CPP funds on October 28, 2008 and repaid it on June 9, 2009.

JPM received $25B in CPP funds on October 28, 2008 and repaid it on June 9, 2009.

Aside from CPP, any bank that had assets affected by TARP purchases -- or by the potential that follow-on programs could affect those assets -- had its balance sheet ballasted in one way or another by government intervention.

All of the banks that survived will say they didn't "need" government intervention, for the same reason that they don't like to borrow from the discount window: it looks desperate to take help from the government. But technically both of those banks received an injection of liquidity via TARP.

As soon as they had a big enough short position and unloaded their long is when the spiral happened. They were the ones setting the swaps prices since they were the biggest market maker for mortgage backed securities.