No, the idea is you just pay for your insurance premiums. When something goes wrong the insurance pays, but your premiums go up. If you're a repeat offender and the insurance company doesn't want to insure you anymore, that's when the dog is taken away from you.
Since this introduces a business, it would be in the businesses best interest to make sure they can insure as many dog owners as they can. If you're hiding your dog then that's your problem. This does give dog owners a benefit, in that they no longer need to take dog incidents into court and defend themselves.Bro, we can't even stop people from getting dogs when they've been legally barred from doing so for animal abuse. What you're saying is firmly in the round of fantasy. People will not pay for the insurance. People will hide their dogs. And when it happens you can go after them for money they don't have and be just as SOL as when uninsured Joe hits you and turns out he's got no insurance and his most valuable property is a pair of pants that don't have holes in it.