I'm pretty sure the rat killing nature of cats is way overstated. Also, in an urban environment where they can much more easily find food thrown in trash, etc., they are unlikely to make an effort to kill rats to obtain that food. Finally, a sufficient cat population that could eradicate a rat problem would likely eradicate the bird population first.
Go read the other responder's article link. Cats can kill rats, but they generally don't bother, and are very inefficient predators of rats. Rats are just too large. Predators don't like to prey on animals that are a significant threat to them. Mice, OTOH, are easy prey for cats and they'll kill tons of them.
There's a group that does this, but supposedly has not made much of an impact on the city's rat population and the practice has stirred some controversy (content
warning: dead rats)
Those with toxoplasmosis are perceived more healthy and attractive, have more lifetime sexual partners[1], take more risk and start their own business more [2] . Seems like a symbiont to me.
"In addition, the documentary looks into the history of cat domestication, and how in the middle ages the church tried to get rid of cats due to their association with witches"
only difference is, now it's "toxoplasmosis and cat scratch fever" instead of witches.
Do you have numbers on the prevalence of those ills? As compared to well-known rat-borne illness?
https://tenor.com/view/killer-upset-kitty-gun-meme-gif-86573...