At one point, AmEx bills showed a recommended date to have a check in the regular USA domestic mail in order for payment to be received by the actual due date, but with most people now paying online or by phone, this seems pointless.

I always assumed that when paying by phone, the confirmation means that one's bank has approved the payment, so that NSF problems should show up at this time and result in one not receiving the email that payment has been received. Of course, this doesn't mean that the payment has been debited from the checking account instantly versus pending status. I think of it as something like an encumbrance for funds in the amount of the payment.