Humans allow cats to breed out of control because they’re stupid.
Humans don't "allow cats to breed". Cats breed whether you want them to or not, often well outside the purvey of people.
What humans have done is to obliterate the rest of the ecosystem. No predatory birds or snakes. No legions of field critters. No native plant life.
Bobcats and pumas were natives of the Texas heartland for hundreds of thousands of years before humans arrived and they got on just fine. Much better at killing wildlife than any Maine Coon, too. You didn't need the SPCA snipping their nutsacks to keep them in check. No more than you needed a breeding program for ground squirrels or passenger pigeons to keep their population size up.