many owners (especially men) grow as attached to their pet's testicles as the dogs do

I'll believe that when I see a man licking his dog.

We looking at this from the wrong point of view.  If there is some guy who's perceived status depends on the dangleosity of his dog's balls, we need to neuter the guy.

It's the end of the world as we know it, and I feel fine

    As a public health/cost issue, I think the effect of this treatment will be minimal, even if it is offered hassle free. Cost aside- and there are those already who are leery at a one time price tag of $250- there's a bigger reason than pride why folks don't get their dogs neutered. Me mum has worked for thirty years as a veterinary assistant, and the top excuse she hears from the owners of male pets is, "well, I won't have to deal with it." I guess the thinking, beyond the dubious ducking of direct responsibility, is that if you own a female dog or cat and don't get it spayed, you get what you deserve when they breed. A double standard? Sure, but it wouldn't be the first time in human history . . .
    It seems to me a better solution to California's pet problem is direct subsidies- with public ad campaigns- to reduce the cost of spaying and neutering. If said subsidies could be projected to reach the cost of, say, $150 million per year*, (which may or may not be possible) the tax nuts could be told their tax burden was going down the same amount, and pet lovers could be sold on all the cats and dogs that wouldn't have to be destroyed. ("Do if for the kittehs!") The only thing slowing this down would be human sloth-- and that's hardly an insignificant factor.

     *Or less. The advantage of a program like this is that the numbers would shrink after the first year.

Ex ignorantia ad sapientiam; e luce ad tenebras

Since there are always stray and abandoned animals being picked up and taken to animal shelters, pets can be had for free (although if you want a pure-bred animal, the cost can be very high).  But once you have your pets, they become expensive.  Pets who become sick or injured can have enormous medical costs, which are much like the costs of treating sick or injured human beings.  Many people have spent thousands or even tens of thousands of dollars on medical care for their pets (which they at originally thought were free).  Still, affluent people can afford such extravagances.  And if there is a new and more expensive form of birth control for pets, some people can afford that as well.  Although in the coming global economic/political/environmental collapse, people will no longer be buying medical care for their pets and instead will be eating them.

But as long as well are living in the golden age of conspicuous consumption, it occurs to me that neuticles may not be the ultimate implant for neutered pets.  Why not super neuticles?  Instead of just installing a lump of silicone, you could install a tracking device that has a satellite uplink, buried inside a lump of silicone, so that you can instantly find out your pet's exact location at any time, no matter where it goes.  And in some glorious cyberpunk future, you might even install an artificial intelligence inside the neuticles, which can talk to you and provide medical information about the state of your pet, as well as talking to other people to tell them who your pet belongs to and where it lives, should the necessity arise.  Exactly what your dog might think about having talking testicles, would be hard to say.

Alternatively, why not use genetic engineering to create dogs that are born sterile?  It's an even more efficient approach.  And they could be born instinctively house-broken, as well.  All sorts of useful tinkering is theoretically possible.