1. There's too many humans, killing humans to reduce the population isn't really a palatable solution to anyone. 2. There's too many cats/dogs, killing them seems much more reasonable to some, and awful to others. They're in the sweet spot between family and wildlife. 3. Pests (deer, rabbits, etc) are over-populating, killing them is pretty much A-OK.
Personally I see a cultural issue akin to wastage. Especially in the US, we love to buy and throw stuff out all the time. I would hypothesize that people who buy/rescue mutts and less desirable breeds are more likely to abandon them after some time. Others who spend thousands of dollars on a specific breed they like will be more likely to keep it the duration of its life.
I see the solution as more of a tax. Stop letting people get cats/dogs for $20. I also don't think inspections are unreasonable. The government already inspects half of the industries, what's one more?