Hunting for sport is forbidden in Judaism. (It is considered animal cruelty.)
Hunting for food (or animal control) is fine, although there are complications regarding having hunted meat remaining kosher, due to what I think are poor rabbinical rulings from Europe when we Jews could not own or work land. Birds are fine.
I think you might have it backwards. I’ve always understood that hunting an animal for food is forbidden since an animal must be properly slaughtered and an animal killed in a hunt would not be kosher. Hunting animals to control population or for their hides or other parts is permissible.
That is one reason why venison (deer meat), a kosher animal because it chews its cud and has cloven hooves, can be eaten but is not because it is not a domesticated animal that can be properly slaughtered.