With deposits in excess of FDIC limits? Yes, absolutely. There are a number of options to manage cash more securely than just putting it all in a single bank.
Either one of 2 things must be true:
1. The FDIC 250k limit is real, and so deposits in excess of that limit must be subject to the risk of failure.
2. The FDIC 250k limit is fiction. If that's the case, then we should be honest and acknowledge that, and price any additional insurance accordingly.
Everyone clamoring for a bailout now wants to have their cake and eat it, too. They want the luxury of pretending like their deposits are insured without paying the premiums.
There are no doubt certain banks that would be more palatable politically to backstop further. I could imagine a national bank with mostly retail depositors and few over-250k accounts is "prettier" politically than if you had a tightly focused regional player that had mostly corporate accounts. I could see politicians saying "Why should Iowa bail out the bad decisions of Bay Area investors? Nobody in Dubuque actually has an account there."
Well, option three is basically what we have now, and my argument is that it's supremely fucked up. As you point out, it highlights how vague, unclear laws or backstops are ripe for corruption and political cronyism.
It also adds a new form of moral hazard: if you're going to blow up, make sure you blow up big. Some small little bank that is pretty contained? Sorry depositors, you're all screwed. But scream "contagion" enough times and use the "nice economy you got there - be a shame if someone were to ruin it" tactic, and get Washington to come to your rescue.
The alternative is scarier: infinite deposit insurance exposes government to larger and larger defaults requiring them to regulate banks down to which industries and loans they can work with or not.
I think the reason such institutions don't exist is that they are very unprofitable compared to regular banks and would be expensive to keep deposits in them.
I mean, if banks offer 5% interest and the fed raises interest to 4%, then there is no problem but if the Fed goes to 6%, it becomes a problem because people switch to the narrow bank. In other words, the banks in the real world cannot compete with banks that take government subsidies, so the subsidy must be carefully chosen instead of being raised arbitrarily.