I agree that the closest America has come to producing its own version of a “warmblood” is the Quarter Horse. The American Saddlebred could also be considered in that vein, although both of course were developed for different purposes than what we know as “European Warmbloods.”
I am long enough in the tooth to remember when AHSA (the precursor to USEF) was run primarily by and for the Saddlebred folks - when the headquarters was in NYC - and the sport horse disciplines were sort of a step child (so to speak) of the old USET. Then USOC decreed that a sport could have only one governing body and USEF was born, which focuses very much on sport horses for the Olympic disciplines. I also think that the old AHSA focus on Saddlebreds was one of the reasons why the QH folks founded their own organization - they wanted more control of rules and regulations that were more conducive to the growth of their breed and their respective “disciplines” than of show and saddle horses.