The FDIC is a quasi-government agency. It is funded by premiums paid by banks but it still is the government.
You have nothing to worry about if you have under $250k in deposits in fewer than 5 qualifying accounts. If your bank closes, you would likely switch to a new bank like First Republic account holders did yesterday or the FDIC would have a check for you within one day. It’s also possible the FDIC could open and operate a bridge bank if no other bank was available in your city or town - they won’t leave a community without a banking option.
Could it be an inconvenience to switch banks? Sure, and we all hope our banks don’t fail but if yours should fail it will all be pretty easy for account holders. Loans are a little more trouble to sort out, but it is very likely your mortgage is held and serviced by another entity, and the mortgage could be in MERS (Mortgage electronic registration service) so MERS would release the mortgage at payoff anyway.
It is more than likely - that’s how the system works. The banks pay the premiums to the FDIC and the FDIC runs like an insurance company with employees, claims, legal and accounting departments. And a LOT more government involvement. When the banks failed in 2008, the FDIC had to get a lot more money from the general budget, so that is what is a little different than other insurance companies, the FDIC has a backup in the US budget.
But that’s how everything works. You buy a stamp to mail a letter and those cents go to fund the post office. You buy a parks pass and it provides funding to the national parks. Those agencies might need a boost from congress and the budget, but the hope is that those using the system (banks, mail, parks) will pay to support them.