I think about that money again when I want to invest it or when I need it to buy something big. Everything else is ATM and credit card, or equivalent stuff. But getting money from a customer or an employer and automatically adding it to a bank account should not become the same thing as investing in the stocks of that bank. Keeping at least a weekly eye on my bank is too much of a burden. The only thing I do is make sure that my balance is well below the limit the state will pay back to account holders is the bank blows up. That's not difficult.
It would be preferable if the regulatory environment would make sure that you as a depositor have an incentive to hold the bank to account.
Yes, you shouldn't have to keep a weekly look on your bank's soundness. You should be able to outsource that burden easily.
Eg with FDIC gone, private deposit insurance would still be allowed. And of course, journalists would also still be allowed to report on banks. Those stories might have a wider audience than now.