I’m happy to report that this is absolutely not what is happening. In fact, merchants aren’t paid directly from the customer’s bank and we can’t decline a collection of money (but can issue a chargeback if the purchase was fraudulent, etc.).
At a high level, what happens for offline payments (and the later finalisation of online payments) is that merchants use an “acquirer” such as their bank, Stripe, iZettle, Square, Worldpay, etc. to send their transactions to the Mastercard network. At several points in the day, these are all processed and a file containing all of those routed to the correct bank in bulk, Mastercard then pays the acquirer who pays the merchant. We’re then effectively left with a large single bill to pay Mastercard for all transactions using Monzo cards that were finalised within that cycle. The file we receive is then processed and the correct amounts deducted from each customer’s account.
I’m not entirely sure where this myth about not being paid has come from but it wouldn’t be anything we have control over. 
As for pubs and bars, holding a card behind the bar as a tab isn’t officially supported or protected by any of the card payment schemes and is in fact actively discouraged. What happens there is they physically hold the card without checking that it’s even active and then try to check the card only after a tab has been run up. 