The FDIC does in fact insure up to 250k per account/bank. You can get more insurance by spreading your deposits between banks, a service some financial institutions will do for you.

https://archive.fdic.gov/view/fdic/4116

It’s not symbolic, it’s literally an insurance company that the banks pay premiums to.

The problem isn't insuring the losses, that they can do. The problem is will you even be able to buy anything after the chaos if a systemically important financial institution goes under? The banks that hold the most ordinary customer deposits aren't like Lehman Brothers and the fallout from one of them going under would be catastrophic on a level far beyond something like 2008.

FDIC was invented for the great depression when banks were not as large or concentrated, nor were they as globally connected and intertwined with day to day business. The reality is that FDIC is far from being sufficient insurance to calm down a collapsing market, that's why we had to do bailouts in 2008, because of what was coming down the road in that regard if the contagion were to spread further.