At least with Platinum and Gold cards from what have seen is long as payment is made before 11:59PM on due date it is counted as "on time". There might be a system generated late fee if payment is made after 8PM mountain time, but then system automatically goes back and issues a credit. Again this is what have seen in past and YMMV. If payment is made at 12 midnight then all bets are off.

If on the rare occasion system does not credit back late fee as above, reaching out to CS may result in them refunding. But don't count on this being automatic, and or the particular rep in question feeling need to go over your account, then sit you down saying 'payments are due before 8PM.....", but as a "one time courtesy....." or whatever. Where there is a recent history of late payments (made actually after due date), then you're on your own, CSR may or may not waive recent fee.

All things considered AMEX is like every other CC issuer; if you are rarely if never late, the one off payment that arrives after due date will usually get any late fee refunded. Especially if payment was made only a day or two late, they chalk it up to "S" happens even with the vest of customers.

Years ago before AMEX moved due/closing dates around IIRC long as payments were in before 11:59 on date due things were fine. This 8PM mountain time seems to be a recent invention that came afterwards.