Knowing that probably answers your other questions.
FDIC says they are.
After receiving a recommendation from the boards of the FDIC and the Federal Reserve, and consulting with the President, Secretary Yellen approved actions enabling the FDIC to complete its resolution of Silicon Valley Bank, Santa Clara, California, in a manner that fully protects all depositors. Depositors will have access to all of their money starting Monday, March 13.
https://home.treasury.gov/news/press-releases/jy1337> The FDIC receives no Congressional appropriations - it is funded by premiums that banks and savings associations pay for deposit insurance coverage. The FDIC insures trillions of dollars of deposits in U.S. banks and thrifts - deposits in virtually every bank and savings association in the country.
https://www.fdic.gov/about/what-we-do/index.html
(If anything, the US govt. and people receive more in taxes from the startup demographic and you should be looking at govt. corruption first if you are keen on saving taxpayer money.)
Why the bail out comment though? FDIC doesn't run on govt money. So even if FDIC helps them out with amounts greater than 250k it won't come out of govt. money.
Of course if the government required FDIC to charter most retail banks, and you had to have a bank account to say not have all your cash stolen by crooked cops by the side of the road in civil forfeiture, then It'd look a lot more like a public good supported by an indirect tax on the return on your deposit. Which isn't far from the truth.