I see and hear this complaint. Cats have the drive and the tools (Google “singing mice”), but not always the skill. Meaning their Mama didnt train them. Some need it.
Cats have a reasonable fear of injury. Rats are tough. Cat claws and teeth aren’t up to abuse. So, knowing how to make the kill in an exact manner throws them for the solution.
Some prefer mice
Some prefer birds.
Rats and squirrels are a whole other class of prey. Hawks, coyotes and even bears are wary of rats. Mice, on the other hand . . . .
Rats are what an entire range of dogs were developed to tackle. The dogs snout protects the eyes and ears. That prey drive is awesome to watch.
I’ve some ranching relatives who talk about the cats they’ve driven far to acquire as it’s from a line known to tackle rats. Motivation, size, training & skill.
The right cat and the right terrier are about the only solution per household pets.
As vermin are a huge problem for vehicles (behind the wall fire in an RV from chewed wiring will total it even if controlled) it’s worth looking into the topic of RODENT PROOF CONSTRUCTION.
Head over to the INTERNET CENTER FOR WILDLIFE DAMAGE MANAGEMENT.
If you will acquire pets who earn their keep, consider that accommodation in your efforts. And the vet bill plus shorter life.
A cornered rat is vicious. Think instead, “ambush” for the predator convenience. How-to-build
..