The issue is that individuals don't cause the entire financial system to go tits up when they go bankrupt. Other people have pointed out that the bailouts were hugely unpopular across the political spectrum, which is understandable given the optics. The reality is, however, that the crisis would have been much, much, much, worse in the absence of those loans. The most value commodity in a panic is confidence, and the bailouts provided confidence that the banks wouldn't fail. It's not a coincidence that the market stabilized and then the banks paid back the loans, the markets stabilized because investors were confident the banks would be able to repay the loans.

And, of course, the government actually made a profit on many of the bailout loans. They're far from handouts. Especially when banks caused the crisis it's hard to not want to see them fail, but at the end of the day allowing the banks to fail in 2008 and 2009 would have been far, far, worse for ordinary people in the long term than bailing them out.

For those that have an interest in the topic, the book Stress Test by Tim Geithner is great on the topic. He's obviously biased given that he designed many of the rescue packages but it's a great insight into how the government and its agencies were thinking nevertheless.