Few years back, I moved to a place in the country where the previous owner fed birds. So there was a bit of a rodent issue. I had with a indoor/outdoor cat, not much of a hunter to look at -- big floofy Maine Coon, and not all that fast. Ran a bit like a bear, thundery, not graceful.

In that first year, I found 23 small rodent bodies inside and out, I got 2 with traps. I found no bird carcasses. Pretty sure he wasn't eating them after seeing one half thrown up on the mat.

So, eh. Some cats kill birds. Some don't. (His brother certainly did. But his brother met something bigger and pointer than he was, and didn't make it to that house.)