FWIW, my employer uses AMEX Travel as well, but this is what happens when I book through them:

1) Reservation is completed with a phone CSR or their online engine. A reservation with both AMEX and US record locators is issued. I get a text-only email with these details. On the US site, i can see the reservation, but not change seat assignments, such as to an exit row that AMEX couldn't access but I can because of status. (Interestingly, I can choose other preferred seats via the AMEX site - the airlines reassign you if you're not eligible.)
2) 20-40 minutes later, I get a PDF of the booking by email once the ticket has been issued. I can fully manage the reservation online.

I haven't had trouble otherwise with either AMEX delays or them getting the FF# on the record. The only time I had a problem with them is when I had to urgently go to the UK last minute for work on a Saturday evening and bought a ticket by phone via AMEX on BA while driving to the sirport, but they had a computer issue delaying the issuing of the ticket, which slightly slowed my check in.