Interesting thread.
I don't pay my AMEX accounts thru the "make a payment function" on their app, but direct from my online bank instead.
FWIW I am based in Canada, and yesterday I paid one of my accounts from the bank website at about 400 PM MST, and this morning the payment has cleared and everything was corectly updated on the AMEX app (outstanding balance, available credit, etc). It's never been delayed for me.