I believe the word you are looking for is extirpated. They didn't go extinct, they just aren't there anymore.
> Don’t we need a grazing species in these areas?
Suppose that depends on your idea of what those areas should be doing.
So the horses that were released (and have thrived!) in north america are originally north american horses. It's part of why they do so well-- they can eat all of the native brush here. Whereas if a cow gets loose in n. america it's gonna poison itself.
https://awionline.org/content/wild-horses-native-north-ameri...
https://returntofreedom.org/new-research-proves-yet-again-th...
That is to say, their natural predators are either not present, not present in large enough numbers, or forbidden by law to hunt them.[0]