Yes, banks as organizations still have the correct incentives. I'm not sure management does, but 1. it's the responsibility of owners to keep management under control and 2. if management doesn't care about investors surely they would care even less about depositors.

They don't want the stock to go to 0, but they don't want to miss potential returns either. When the FDIC will cover all depositors, investors (from a purely capitalistic perspective) would demand to use as much of their capital as possible to maximize returns. As we've seen numerous times before in the past, a large quantity of investors is myopic and regulatory oversight often comes too late or after the fact.

Then why is (was?) there a limit of 250k anyway?

There has been, over the nearly century we’ve had the institution, many many waves of feelings regarding it. That said, three prominent ones: the amount one needs to reserve, and therefore the rates one needs to charge banks to build the reserve, are sensitive to what portion of the sector is actually insured. There exists a sense that rich, sophisticated people and entities can arrange for their own risk management at their own expense rather than a society-wide program with implicit government backstopping. Finally, there is frequently strong disinterest in either the reality or the appearance of public funds being used to backstop the banking industry.

And so that is the reason for the limit. It gets bumped up every few years, partially due to inflation and partially due to the increasing wealth of the upper middle class and retirees, who the insurance fund is primarily aimed at motivating. It would not be effective in its aims if “local elites” at the typical community bank felt like they still shouldered run risk, and local elites in 2023 are substantially wealthier than they were in 1945.

To guarantee the FDIC itself doesn't fail.

Does it matter in regards to what I said?

To prevent bank runs.