Computers, on the other hand, are perfectly happy to use an expired timer to generate mail, a dashboard icon, or any other shout for attention that you please -- as long as you actually set it up. Paying attention to it is still a human problem.

I'd say you're dead right -- this is the only safe assumption. And any email client smart enough to check incoming email against a regexp could be extended to raise a not-easily-ignored alert if no pattern match (indicating success in processing a payment) had arrived by some chosen date of the month. Yet I've never come across such a feature in Gmail, Fastmail, nor Thunderbird. Maybe someone sometime has hacked such a thing into Gnus ...