Further, bank software makes a distinction between different financial instruments, for instance, if you deposit $8K in cash, or in cheque, or in wire transfer the deposit information will capture that difference.
Even $10 billion in 100s, per year, divided by say 200 days that a bank is open, is $50 million per day. That is, 500,000 x 100 dollar bills, being deposited and counted, each and every day the bank is open.
Does it sound reasonable to you, that their software wouldn't record that, and it would seem shall we say "odd" with even minimal review of their bank branches? It would take multiple tellers an entire shift to count such money.
EDIT to add: even counting 1000 bills per minute with high speed counters, you have 500 minutes, which is 20 minutes more than an 8 hour shift with no break.