Modern domestic cat is a poor rat catcher because their role is to be a companion
Compared to 2,000 years ago the modern domestic cat is smaller, pampered and flabby and not, in general (there will be exceptions), an effective rat catcher. As prey mice are safer and smaller.
Modern domestic cats are 15% smaller than domestic cats of Ancient Egypt. The cats of 2,000 years ago were not only larger but they were all striped tabbies and they were kept as rat controllers. In fact they were 10% larger than their wild cat ancestor so they must have been selectively bred informally I’d suggest.
The conclusion is that 2,000 years ago the primarily role of the domestic cat was as a pest controller. They were working cats mainly with a secondary role as a companion. Cat owners in those days selected cats for their size and pest controlling skills.
For instance a typically large male domestic cat was buried on the Red Sea coat 1,800 years ago had the remains of five black rats inside his stomach and one more lower down in his gut. This cat was a prized rat catcher and buried with honours.
The African-Asian or North African wildcat (the wild version of the domestic cat) preys on animals that the modern domestic cat would shun and might run from such as rats, cobras, vipers, domestic lambs, young antelope and so on.
The fact of the matter is that over the last 1,800 years since that super rat catcher was buried, the domestic cat has evolved into a home loving companion often kept indoors all the time and pampered. They are fed commercial food and although their desire to hunt is there still, of course, mice are much easier to catch and eat at 30 grams compared to the 300 gram rat. It is about taking the easy prey which is common sense. In general, the modern domestic cat is soft and flabby (literally) compared to the ancient variety.
This is why domestic cats prefer mice and reptiles (when present) to birds. They are easier to catch unless the bird is dead or dying. And the modern domestic cat is not always keen to hunt. My late lady cat only caught one mouse her entire life and she did not eat it.
There are individual cats today who are great hunters like my current male tabby but in general and compared to 2,000 years ago today’s domestic cat is not good at rat catching.
That said rats are very aware of the presence of domestic and feral cats and disappear in their presence. The modern domestic cat is a rat deterrent but no longer a great rat killer. If you want to get rid of the presence of rats, the domestic cat will do it. The rat will hide but not be destroyed.
Sources: Myself backed up by Cat Sense written by Dr Bradshaw and researchers led by Fordham University’s Michael Parsons in study (a limited study however).