I can't stand most outside cats either. I did have one who left birds alone but loved hunting the ground squirrels we had in AZ. I tried keeping her in but she was very adept at darting past legs, especially if my hands were full. My current cat Monkey has never been outside except one time when she was rubbing against the window screen and it ripped. Scared her to death and it took me 45 min of chasing her to catch her. She won't even go out the door if it's propped open.
Here are a few cat-bird gifs I thought I'd share. I love the 1st one.