> instead of just not breeding more

As one might guess, the "just" there is doing a lot of work.

Common pets reproduce just fine without our intervention - in practice it requires costly maintenance programs just to keep their population under control (not to mention the massive and ongoing eugenics program required to produce only the desirable breeds/attributes).

If your neighbourhood isn't overrun with stray cats and dogs, that's because someone is funding a spay/neuter program, and someone is funding animal control to round up (and ultimately euthanise) the rest.

All animal husbandry systems have to be maintained via regular culling, and "eliminating" pets would require orders of magnitude more killing