I bred to a SF stallion back in the '90s who was also approved NAWPN.
I also bred to First Class twice who was in the Selle Francais stallion directory since he had French blood. First Class, BTW, won the Bundechampionship. At that time, I didn’t know the significance.
Anyway, I was just learning about warmbloods so I studied everything I could get my hands on.
AA is a distinct French breed. Matcho is one of the landmark stallions used in WB breeding among some other SF.
In the USA, the Anglo-Arab is a cross of an Arabian to a TB. It was pretty popular in the 80s when I first got into horses. It faded away as the WBs came in and riders wanted a less explosive ride.
I think a domestic AA might have a place if the parents were inspected and evaluated for character, rideability, etc., as all the warmbloods are. But it would be seen as an F1 product, the downside being the absence of a career track in the Olympics sports for generations.
The AA in France would have that career track. Again, look at the bio page on Baladin. Lots of info on his close relatives in pertinent competition.