These are not “wild animals”, they are feral. They are descendants of domesticated animals that have established populations in environments where they do not belong. Think of the problem with cats; this is the same.
Feral horses and burros are invasive species in North America that have massive harmful impacts on ecosystems. Populations need to be managed and culled. Sentimentality results in animals being kept in holding facilities rather than euthanized, opening the door for problems like these.