have had lots of cats that roam free (like forty plus barn cats once upon a time) and never had one that got birds except one cat and then only if they got on the porch or something. had to be a trapped bird or slow bird.
not sure I'd worry about the birds. I know it's the big thing - cats kill song birds! oh my! but mine were mousers not birders so maybe it is a learning thing when young.
if you are feeding birds, and have a cat, try to control when he goes out maybe - they tend to feed more at certain times yes? otherwise yep, you're just like a deer hunter putting out corn
build a large cat pen! then the cat is safe and the birds are too. mine like their cage, when the door is open they are often in it.