People are picking holes in the OP's various proposed solutions to the problem, but IMO the issue isn't really whether OP's specific solutions would work — actually, I think OP probably couldn't care less about those specific things happening or not, what she wants is to not be forced to pay the cost of other people's pleasure.
Although there's a certain amount of tolerance of other people living their lives that's necessary for people to live in close proximity — maybe some barbecue smoke in summer, or ordinary noises of living, or seeing well-behaved dogs out and about — and although there's usually a certain amount of willingness in communities as a whole to pay for certain costs or detriments that are caused or incurred by some more than others, in general there's an expectation that people shouldn't do things that benefit themselves to the detriment of others. When someone wants to do something which could cause a problem for others, it's usually considered to be their responsibility to find a solution and pay the costs themselves, either individually, or as a group, through taxes, etc.
With pet dogs and cats, the people who buy these animals purely for their own pleasure and recreation often use the fact that they are animals as a justification for the fact that it can come at the cost of their neighbours' quality of life. Other people have to deal with dangerous and disgusting animal waste, and spend their own time and money on installing deterrents, doing cleanup and/or obtaining replacements for damaged property. Regardless of whatever fox shit/pigeon shit/hedgehog shit may be produced by local wildlife, cats and dogs produce copious additional shit, usually in locations far more disruptive to humans, and purely because someone wanted to enjoy owning that animal. They're not wild or native, and they're there because they're owned by someone.
I think what OP would like is for owning a dog to be like a lot of other human recreational choices that can harm others — the ideal being that no other person is paying part of the cost of the dog owner's pleasure. If you live next door to me and own a dog or a cat, I should not be able to identify any way in which I'm harmed by your choice, just as I would expect not to be harmed by the fact you own a pet snake or enjoy playing in thrash metal bands. Disruptive barking all day, scratched-up seedbeds, shit on the pavement, shit on shoes and wheels, the smell of piss on my fence, toxoplasmosis, apprehension of being knocked over or attacked, actual attacks, muddy pawprints on my coat, dog snot and slobber on me, all of it is me paying the price for someone else's pleasure.
I think that to OP it wouldn't matter how exactly it happened, how dog owners became responsible for ensuring that some of the costs of dog ownership weren't being distributed to other members of the community — whether it was through all dog owners suddenly becoming considerate and responsible, or enforcement paid for through licences or taxes or fines.
I love dogs, though can't have one in my current circumstances, but too many dog owners don't care that their dog makes other people's lives a little bit worse.