I will answer and you can answer this question:

Why is the government involved beyond that limit in this case? Could it be they want to give the average person peace of mind while retaining the flexibility to handle a restructuring however they deem best?

Because the bank continues to have assets, and those assets will be dispersed to cover depositors.

It's because there is a cost associated with this insurance. It's not a bottomless pit of money, as FDIC pays out using the funds it collects from the participant banks and receives no funding from the government (https://en.wikipedia.org/wiki/Federal_Deposit_Insurance_Corp...). I think it should be possible for the bank to get better coverage, but they should be paying much larger premium if they want insurance up to 250mm instead of 250k.

I also think that similar to "FDIC insured" labels in bank branches, FDIC should require posting "13% of deposits are FDIC insured" to help assess risk for those clients that have uninsured funds.