I think the general consensus is that it's best to book flights directly with the airline, especially if you have status or want protection during IROPS.

It's not necessarily more expensive to book airfare on Amex Travel than directly with the airline. I believe Amex Travel doesn't charge or waives booking fees for Platinum and Centurion cardholders. People who see different prices between the two often aren't making apples-to-apples comparisons between fare classes or might be comparing a basic economy and a non-basic economy or non-restricted fare on Amex Travel (or another online booking portal).

Plus, Amex Travel has a discount for international premium airfares (first, business, premium economy) with participating airlines, which could be cheaper than booking direct. (See: USA International Airline Program (IAP) relaunched as discount benefit (2017).)

Sometimes Amex Travel has higher hotel rates because they're selling refundable or more flexible bookings. The direct option might be cheaper because of discounted rates for loyalty members, refund or cancellation restrictions, etc., but, it's generally a better idea to book hotels directly so that you can earn the points or elite qualifying nights or stays anyway. Also, you should generally be able to earn points or elite qualifying nights when you book through Fine Hotels and Resorts (see: Report chain hotel rewards or benefits on PREPAID FHR and THC stays) and sometimes The Hotel Collection.

I get a lot of value from the airline credit (United Travel Bank or award ticket taxes), Uber credits and FHR bookings. I am almost always able to check in early at 12 p.m. if not sooner (technically, it's subject to availability), and the 4 p.m. guaranteed checkout is incredibly useful when I need it because I have an evening or late-night flight.