I grew up rurally, and we had cats as pets not to control the rodents.
Now that I live in suburbia, my cats are indoors, but I have had cats before who had outdoor access.
The miserable tuxedo stray who couldn't mew when he came to our door aged about 8 and lived to be over 20, literally never caught anything. The birds in the garden that took flight when another cat came in actually ignored him.
The gorgeous ginger stray who rocked up onto our windowsill with the biggest set I have ever seen on a cat (we soon had the vet see to that!) cornered a rabbit once with his partner in crime (of whom more later!) but didn't do anything to it and they were both ignominiously grabbed by the scruff of their necks and removed from the situation! He brought us an empty cigarette packet, a cheese wrapper and a used teabag.
Our beautiful blue tabby, who we inherited when his hateful owners moved house and left him behind, was a partner in the above-mentioned rabbit scenario. He also cornered a baby robin once, and we took it to a vet who took in waifs and strays but it sadly died. He came home a couple of times with a nice freshly cooked chicken breast, which he devoured on our patio, and another time with a pork chop. I knew he was inclined to go in neighbours' open windows but I never found out where he got the lovely food! I remember one Christmas Day when he sat on the windowsill for hours because he could smell the turkey cooking, but when I passed him some out the window, he fell off the windowsill in his excitement and couldn't find the turkey!