I’m sure they’re already against breeding. They’re just saying the root cause of this problem is domestication. Whether it was for “pets” (as in just for cuteness) or for labor like you say.

If every domesticated/not wild animals were captured, neutered/spayed, and adopted or better, taken care of at in a centralized, public fashion of animal sanctuaries, these domesticated animals will in theory go extinct. And for some “pet lovers” that’s hard to accept.