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Cats are vermin. I love having one around, but they will eventually piss in your heating ducts.
My family has 3 cats, my buddy and I have 2 Bengals, and my girlfriend has 1 large dumb boi.
While the house is stinky in the years I have lived on this planet. I have never had the issue you have described and I would never describe any of the cats I know and love as vermin.
The thing is, they don't care. Eventually it will be too much effort for them to go to the litter box, and they'll piss on the couch, or find a spot in the carpet that they sprinkle for weeks straight until you get fed up with the smell and try to locate it. Then, you can't get the smell out, and they key into it as a new pissing spot because it smells like piss. No matter how hard you try, they just keep pissing there. You hit it with hydrogen peroxide, trying to get the smell out so they don't think it's a pissing place, but you can't, and every time they piss it renews that scent. Then they get diabetes because they're old, and they piss every 30 minutes, and you can't keep up to the pissing. They find a new spot, and you can't find it because everything smells like piss. They piss on your clothes, in your shoes, behind the stairs, and eventually, in the heating vents where the warm air releases new piss smells throughout the house, so everything smells like piss. And that's the worst place to piss, because it rolls back down the ducts where you can't get to it to clean it up. And when that cat dies, or you have it euthed, the next cat thinks "Hey, this whole house is a place to piss" and then it starts all over again, only faster this time.
So. Anyways. Cats are vermin.
Sounds like you probably shouldn't have animals.
Sounds like you probably never have. Or not for long enough to know what I'm talking about yet.