Because pets and stray pets are an ecological disaster, especially if they freely reproduce in the wild.
It's fine if you keep them at home, but not if you let them roam free. cats will kill anything they come across just for fun. entire species and ecosystems have been wiped out because of this.
Our neighbor had freely reproducing barn cats that he didn't care about at all, so you'd have sickly cats all over the neighborhood, regularly finding dead ones. It isn't fun for them either. It's best for everyone involved to leave their reproduction to breeders.
You had me until your last sentence. Breeders aren't a guarantee that it will be done properly.
Mixed breed or mongrel dogs are sometimes healthier than purebred dogs that are often inbred. Same goes for purebred cat breeds, if they are overbred with close relatives you will get bad genetic issues.
Breeders aren't saints. They make money by breeding cats that are in demand, even if that means breeding disfigured cats with (too) short skulls because they look SO cute : /