We don't take merchant deposits through the drive thru.
Considering the potential time involved in verifying a business deposit with cash, I would have assumed that no one did. All three of the banks where I worked had signs on the drive thru telling commercial custoemrs to come inside...
As far as allowing "drop and run," not a chance, not a snowball's chance. Those tellers who are "saving time" are the ones who will be under suspicion if the getaway driver does not wait until the cash portion of the deposit is verified and there is a discrepancy. The deposit would be placed under dual control (just like the night drop) and the business would be notified to send someone back.
From my perspective, the point is moot, but, yes, you would need the getaway driver's ID if you allowed the practice.