I think most people get "moggy" cats, because that is what most people have, and they tend to self-reproduce.
Moggy cats are on average, healthier (because they arent inbred, or have serious renal or respiratory problems that "purebred" cats have), and smarter than their "show" counterparts.
They also tend to be "free to a good home" because mittens managed to get out and find some random tom somewhere and then popped out 12 kittens a few months later, and her peoples cant afford to keep 13 cats.
I think spay and neuter are good control methods, but introducing a genetic nuclear option is over the top. Nature is significantly less kind about things than humans, and she is more than willing to murder them all, so to speak. Currently, stray cats live terrible, bleak lives-- but no worse so than any other wild predator of their size.
If you are a pet owner, keeping your pet from replicating out of control is part of being a responsible one.
I personally see the market for cats go way up if there is an introduction of a nuclear population depletion option in the population, because the depression on the market from ubiquitous availability of moggy cats will vanish. I dont think the benefits would outweigh the costs, and think it is dangerous to do so.
Now, if you want to totally destroy a population that should not be someplace-- like mice in NZ, or cane toads in AU--- by all means! Nuclear Option Ahoy!
Companion animals? Better to institute more aggressive fines and fees for irresponsible pet ownership.