We could use Tokkay Gekkos to get them. Then the cockroach problem becomes a gekko problem. Then we use a cat and the gekko problem becomes a cat problem.

https://en.wikipedia.org/wiki/Tokay_gecko

No, that's the beautiful part. When winter rolls around, the cats simply freeze to death.

Since Skinner recommended gorillas: https://youtu.be/P9yruQM1ggc

... I had to check [0]: at least mountain gorillas can survive temperatures below freezing.

[0] https://www.worldwildlife.org/species/mountain-gorilla

To be followed by a similar article, titled "Cats becoming resistant to most winters"

Is having too many cats a problem though? Looks more like a bonus to me

Yes it is; I think you're thinking about healthy, pet cats. I have a couple, I've also worked with the local cat population control group on the pro-cat side.

Too many cat's is a huge problem. They carry unique diseases, most notably those transmissible to your pet cats. Also, toxoplasmosis gandhi which will kill large mammals in an area, including whales. It also infects humans through under-cooked meat. Cat scratch fever. Cats can also transmit whatever local diseases and parasites other mammals carry, but effectively as they tend to live in large colonies close to humans.

Cats murder your local bird populations, which in my area, includes a number of endangered species. I'm sure in other areas, there might be other endangered creatures cats remove.

And finally, there's the typical nuisances of getting into food stores and trash, dying on motorways, feces accumulation (even before considering diseases), and noise complaints.

Further a large feral cat population creates rabid, cat-hating humans which IMHO is also an issue.

To many cats results in no birds, resulting in more flying insects