They aren’t just hanging out. They were driven to extinction by the early 1900s and repopulated into the wild by zoo breeding programs. It’s a textbook bottleneck genetics scenario and very nearly failed had then not found 1-2 wild horses to increase genetic diversity in the 1960s.

Although their population has been increasing successfully, the biggest reason it is working is because they live in areas of Mongolia that are not heavily populated by humans. For the counter example, watch Montana get rid of its re-introduced wolf population. Humans love animals in so far as they don’t inconvenience us. But our societies will happily get rid of animals that get too close, even though we’ve left wildlife with so few places to go.