The Equus genus actually evolved in North America before spreading across the land bridge into Eurasia. The North American horses went extinct around 12000 BC likely due to climate change turning the steppes into tundra (and possibly due to human hunting pressures.) Eurasian horses didn't have it so easy either, I think. They were rare for a while before bouncing back.
If Przewalksi's horses crossed back into North America, they'd have to contend with the same changing climates and human hunting pressures that were driving other big animals extinct. Eurasians began domesticating the horse around 4000 BC. As the land bridge is closed by 11000 BC, those horses would have to survive at least around 7000 years before humans realize their worth. If some herds are lucky enough to exist in relict populations that survived the initial wipe-out, I suppose they could be domesticated, but it would probably take centuries for their genetic diversity to go up after that kind of population loss.
Even if the Native Americans never figure out riding, they have a steady supply of meat, leather (and possibly milk) that can be herded like sheep, goats, or cattle, so you'll at least have larger populations based around a reliable food source. Maybe they'll try burning forests to create more pastureland for their herds and hunt down animals competing for the same food source.
Being around horses in denser populations will result in more diseases developing and thereby more robust immune systems in survivors. So, at the very least, Europeans will discover a more populated America not so easily conquered by virgin epidemics.