Sabre used to be difficult with giving production access unless your website has a certain size, not sure if that has changed.

Amadeus is pretty good as an API, but expensive and misses AA/BA fares (no idea what other airlines might be missing).

KIWI is a pretty good (and free) API, though as you rightly pointed out it does not tick the fare class box. Also, some of the fares (especially premium cabins) are stale, this only becomes clear when you move to the actual booking stage.

There are some airline specific APIs (Lufthansa is good), but the offering is very niche for the airline itself or related airlines.

If you find anything better, would be interested to hear about it.