I’m sure big city banks do, but out here tiny town fly over country, I’ve seen the put the rolls from the counting machine into their cash drawers.
This may be why big city banks charge counting fees, because they are paying shipping on unrolled coins to the fed and on rolled coins back from the fed.
Tiny town banks aren’t that dumb. I only bank at credit unions and small locally owned banks because they still seem to have common sense.
I'm sure they do that in small towns like you said. However here in DFW we have seen the same coin shortages yet we have a fed reserve bank in Dallas. Here if you ask for a couple boxes of nickles they will usually come directly from the fed reserve bank. Yet there is a supposed shortage of coins. I believe the shortage is due to the .gov just not doing their job and causing this shortage deliberately. This forces people to use their traceable plastic.
It's no secret the .gov wants to get rid of cash transactions and has been trying to implement ways of tracking them for some time. They really want cash to go away so you can't EVER be anonymous.