Getting cash and service after a disaster is, of course, a concern and the actual outcome remains to be seen. We only find out for sure in the moment. Hard to know for sure whether a large or small bank gives better service ahead of time.
But I must point out that to a large bank you're likely to be so small a depositor, excluding the resident millionaires here, that your little more than an annoyance. At most your an account number and someone to nickel and dime with fees for extravagances like ... well ... walking into a bank, talking to a teller and handling your own money.
Chaps my rump when they act like they are doing me a favor allowing me to deposit and manipulate my money. Looking down their noses at me and like I was a Dickensian orphan asking for more porridge ... and bill me for the privilege?
On the other hand at the local credit union I'm a name and a face in addition to an account number. I have met the president and many of the officers and wasn't made to feel like a Dickensian orphan reaching above his station. I know where the president lives, a modest house and the only one she owns, and it is just a few miles from my house. I can't know for sure how it goes down after a major disaster but I expect that being a face and a name, and their habit of working primarily with small depositors, I get a little more attention and consideration.
I also feel like I'm more in charge of my money. Which means a lot.