When Silicon Valley Bank imploded after a bank run, the FDIC stepped in. But the majority of SVB's deposits (over 80%) were above the $250k FDIC insurance limit, and would not be legally required to be covered.
The Biden administration quickly announced that these deposits would also be covered. This made sense. Without those funds, a lot of companies would not have been able to make payroll in the following weeks, which would have been hard on a lot of employees and local economies. There would have been more bank runs as companies scrambled to move their excess deposits out of any bank that didn't seem rock solid, or "too big to fail", which would have meant more smaller banks would fail.
But there is a risk of moral hazard, in which the government safety net encourages risky behavior. Put another way, capitalists get to keep all the gains, and socialize the losses.
The moral hazard doesn't really occur for the bank owners themselves, because their equity is wiped out.
The problem occurs with the depositors, who can search out the higher returns offered by banks which are desperate for deposits. This is what happened during the S&L crisis some decades ago: banks such as Corus Bank (Chicago area) were offering much higher CD rates as they went into their slow death spiral. I was happy to take advantage of this for my IRA investments. I was reading a lot of reports that Corus was in trouble, but I didn't care. My deposits, of course, were well below the FDIC maximum.
How to avoid this problem with large depositors? The FDIC should guarantee only the principal, not any interest (for example, any interest earned in the past two years). This would substantially lower the incentive for interest rate shopping by large depositors.
In order to try to control gaming the system (e.g. frequently moving funds from one bank to another, to restart the interest earned), the FDIC would not cover any interest earned in any FDIC institution during that same period. So, if Bank1 fails, your principal at Bank1 would be covered, less any interest earned at Bank2, Bank3, etc. during that same lookback period, if the money in Bank1 had been deposited during that lookback period. This, in fact, provides an incentive NOT to move money around.
This is simpler than it sounds. The main requirement is that financial institutions change their 1099 reporting to split out interest subject to FDIC coverage from interest not subject to FDIC coverage. This isn't complex. Fidelity, for example, already tells me how much interest is from US government bonds, how much is tax exempt, and how much is tax exempt by state, and so on. And when I check their fixed income information, I can see whether what I am about to buy is covered by FDIC insurance.
A bit of personal history
No, I'm not old enough to remember the bank runs of the early 1930s!
But my parents and grandparents had CDs in the 1960s from Southern Savings and Loan in Louisville, KY. It looked like a regular savings and loan, but it paid rates that were a bit more attractive. It failed. Turned out it was not FDIC insured; it may have been insured by a state fund that didn't have enough money; I was a kid and don't remember all the details. My family had moved from Kentucky by that time, but still had some money left there, perhaps in longer term CDs, or CDs that had be left to automatically renew since the interest rates were more attractive.
Eventually, years later, they got most of their money back in a series of payments. Evidently the savings and loan had assets that covered most of the deposits (probably 20 year mortgages that were gradually paid off). We were lucky enough not to need the money in a hurry.