The point is that banks would incur costs from protecting themselves against a spate of runs on the banks and if it's true that banks pass on all their costs then those costs would be passed on to their customers.

Pretty much - although like underwriting natural disasters claims tend to cluster together.

It is an insurance fund; for every $100 in deposits, banks pay a few cents in insurance premiums.

notch898a is referring to how, when the full backstop of deposits regardless of size at SVB and Signature bank was announced, the government said that a) it wasn't a bailout because taxpayers wouldn't pay anything, and b) there would if necessary be a "special assessment" (i.e., mandatory additional premiums) from all banks. Our point is that a) is nonsensical given b), because of the total overlap between US taxpayers and US bank customers.

If things get bad, it will get funded by taxpayers. It's definitely "too big to fail".