The events of the bank closing may be deflationary.

The giving of now-gone money to the depositors is inflationary.

The fallacy of your above statement is you're considering the system of 'fail, then FDIC pays out depositors' when in fact the alternative is 'fail, depositors eat losses'. The former is inflationary relative to the latter, and in fact punitive to those who chose banks that didn't fail. The net difference between the two is the discussed inflationary event 'FDIC pays out depositors.'

No, it doesn't increase the money supply because the FDIC has a pre-existing fund for it. They're not funded by the Treasury and when they are they pay it back.

Though the US could've used more inflation at any point up to 2021 considering our inability to ever get unemployment low enough. (since they're theoretically more or less directly related)

Pre-existing Funds that are just sitting waiting around to be put into depositor accounts once a bank fails have low to no velocity. Giving them to money-spending depositors is inflationary by virtue of increasing the velocity of that money that was previously just sitting on the sideline. Even in the scenario you present is inflationary.

Eh, if it was sitting in a bank before and now it’s sitting in a different bank that’s not a difference. The depositors never got an unexpected increase in spending power, they just avoided an unexpected decrease.

What matters is where the old funds went. The bank might’ve been spending it out the back, but in that case the inflationary actions already happened.