It baffles me that US payment systems are so archaic that this is even possible.

Like, where I'm from, cheques literally don't even exist anymore - banks have all stopped issuing them. It's all electronic, with 7-day-per-week payments between any bank in the country, and same-day payments within the same bank. I can send my GF money and she has it an hour later (without even needing a 3rd party app or anything - straight to her account).