Getting back to the topic of AMEX travel as an OTA, my old employer booked with Amex travel through Concur and the fares were often higher vs. booking direct. It would not show the lowest non-basic economy fare code that was bookable, even for simple Point A to Point B flights with no layovers. Then you paid an additional travel booking fee for the privilege...

So they definitely do that and I bet they make so much more with corporate bookings where you are forced to book through the portal vs. doing it by yourself.