The banks primarily invest in mortgages, and treasuries, and so a lesser extent business loans. That's about as 'safe' as it gets! If they wanted inflation protection they could invest in, what? Securities? Then they'd be an investment bank. Even if this was legal, then if market went down and there was a run and there would be an article about how the bank should have held bonds and treasuries. If they held cash they wouldn't be a 'bank' anymore and couldn't offer what few services they do.
Banks are exposed to interest rate risk. They borrow short and lend long. Anybody taking a look at their checking account today will see they can get higher yields by moving their money around from their traditional bank to new high yield accounts from e.g. Synchrony and many others.
Based on this I don't see this ending with First Republic.
Now, I am sympathetic to claims of mismanagement, but I don't see it at the bank level. I'm just a amateur here but if I had to point my finger at something that exacerbates these problems its the US government's continued support for long (15yr+) fixed-rate loans. These really appear to be US government creation.
Meanwhile our federal gov is running record peacetime deficits. If interest rates remain high (and its hard to see them coming down very quickly) its going to blow a gaping hole in the federal budget.