That last paragraph is fascinating! How did you react to their request to pay cash? How did you count the money to ensure that it was all there - did you sit there and do it when they arrived, or employ someone or did you visit the bank together?
You weigh the cash with kitchen scales and check it's not paper blanks. That's about it. You have already exchanged contact details and IDs etc so it's all above board plus there is a solicitor present to hand over keys and act as immediate escrow. If there is a miscount of a few notes, ignore it and put it down to an accounting error.
Then you go down the bank, tell them you have a ton of cash to pay in, hand it over and it arrives in your account instantly. You have to provide ID and a receipt for the cash to say where it came from and that is it.
The scary bit was parking up in the local high street with a cash full of cash in your hand at a parking meter paying, then walking to the bank. When you have that sort of cash on you, you look like you have that sort of cash or a dead body in the case. Bank didn't even really think it was surprising.
Pretty depressingly boring.
I watched helicopters arriving in Hatton Garden from an office I worked in once and they were loaded with expensive diamonds and stuff with men in black suits - this was nothing even as remotely interesting or glamorous. The suitacase had no handcuff and was a bog standard samsonite one with dents all over it and the guy who bought it was a retired golf supplies salesperson.
A $1000 bundle of 20's is 50 bills. Usually they're wrapped up like that. That's only 15stacks. You'd only really have to count one of the stacks, if you didn't trust them.
In the US, at least, there are money counting machines in, well, the two bank branches I've been using for a decade. Different banks, one small, one medium of size, in "flyover country" (Joplin, MO). The tellers don't entirely trust them, but they're at minimum useful in getting a double check of amounts. And quite fast, > 10 bills/second.