Because it minimizes harm and stabilizes the economy. There are benefits to providing safety nets to people who didn’t have a hand in their own misfortune. Specifically those people get to continue providing value. The modern economy is not zero sum. Work creates value, it is in everyone’s interest to stabilize productive workers.
The owners of the bank aren’t being bailed out. The FDIC is only paying back depositors. That is the working class. This bank doesn’t just cater to big bad VCs.
When depositors lose their savings through no fault of their own the whole system comes crashing down. That won’t help blue collar workers.