I echo what @dmveventer said and also:
…overwhelmingly, Western-oriented sport is much bigger than English-oriented sport, especially in the wide open places in the US where horsekeeping is not limited to the very wealthy. As such, nationally our “default” horse is the Quarter Horse.
There’s technically nothing stopping someone from building a foundation warmblood herd out in, say, Iowa and offering cheaper horses for English sports. It wont happen because those horses still go for $$$ so why would anyone dilute their own market? Anything remotely warmblood is still going for low 5s. Esp if it’s got a brand on it, it will sell…old, lame, unsafe, whatever.
Our other popular horse sport is racing, although that is obviously a spectator sport, but it means the other available breed is the thoroughbred.
I personally believe our thoroughbreds suffer a lot of bias and are very underutilized. This is beginning to change, but they are an exception at the top levels of “polished” pursuits, where the form is as, important as the effort, even if they are more widely seen in sports like eventing where raw athleticism is much more important.
Think the debate in figure skating, where “artistry” is a very subjective and amorphous quality, but vital to winning.
“Hunters” has always been oriented towards a pursuit by and for the wealthy, even though all but the most downhill-bred QH can athletically handle popping over a 2’6” course. Which is what 90%+ of what anyone is actually doing.
Any suitably built horse, thoroughbred, appendix, warmblood or otherwise, that can jump the required height could hypothetically do what the OP seeks to do but the degree of polish that is the barrier to entry in this universe is way above the baseline ability to safely get over the jumps.
And that is because of the coastal money that props up English sports, and the aesthetic culture that sprung up around the money, as well as our unique history as a “cowboy nation” in the rest of the States.