Banks get their coins through deliveries from the local federal reserve banks. Order some coins from your bank and they will come in cardboard boxes just as the come from the fed reserve.
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.