i don't think there's a dog overpopulation.
i do believe there is a cat one. at least around here, kitten season is overwhelming. there are so many stray and feral cats reproducing and it's out of control. there is low cost TNR available but i'm not sure how accessible it really is, and for the most part, these aren't "owned" cats, and people aren't really that excited to leap up and take responsibility for them.
but dogs? puppy mill business is BOOMING. byber business, petstore business. they are selling puppies and puppies and more puppies. logic tells me that if there were truly an overpopulation problem, this wouldn't be the case. there wouldn't be demand.
dogs languish in our shelter system not because there aren't homes available, but because we aren't doing a very good job of getting dogs into the homes. for a bunch of reasons including but not limited to: lack of marketing, difficult adoption requirements, lack of hours, the myth of the "damaged goods" shelter dog, etc. in truth, the number of dogs killed in shelters has dramatically DECREASED over the past several decades. but you would never guess that from listening to the talking heads.