I was the bird-rescuer last year. Never my cat though - always others and I was the one to try and bring them back to life! A seagull had been injured (hit by a car I assume as no obvious marks from cat attack) and I watched it all day, willing it to either fly of or die peacefully - neither happened so I called the RSPCA, give them their due they came and got it. I reckon they will have PTS and in fairness, I can see why - the money it would cost to fix it vs the trauma of putting the bird through that rigmarole.
Last summer I only had Rosie (got Bert mid August), and she tends to favour Moths and mice. She never kills, always maims 
So we were sitting watching tv, back door open and next minute the dog went ape-shit, petrified, running away - sure enough Rosie had a moth the size of my hand (I swear to you, it was huuuuge) with one wing not working but flapping like a good-un! Rosie drops it at my feet (cue me screaming the house down at giant-flappy-moth-beast).
Wasn't sure what to expect with Bert, considering he's 9 months old and has a dodgy pelvis (after being fractured aged 5ish months). At the very least (considering he has a shiny collar with a bell) I expected a sparrow. I was greeted with a Magpie on Wednesday. Thankfully dead. I looked at the bird, and then at (a very content) Bert and there wasn't much difference in size! He then proceeded to lie in the garden, on his back, in the sun and make the most ridiculous noises at every bird that flew by (a high pitched meow/growl/purr??) as if to say "yeah, you know where it's at" 
I take no joy in my cats bringing me presents, but if I didn't see the lighthearted side of it, I'd sit here crying!