I adopted two slightly older cats from rescue last year. Both were used to being outdoors but have shown no inclination to go out since they've been here. One of them loves catching flies, moths and spiders, and watching birds out the window.
We've had 5 previous cats since we got married (had cats at 'home' before that for a lifetime!). Two we got as kittens and they were indoors babies. The other three had no interest in hunting. One brought home a cigarette packet, a cheese wrapper and a teabag in the 9 years we had him. He did tag team once with another of our cats to corner a rabbit, but I grabbed the little buggers before any damage was done!
That other cat once caught a young robin. That was it in 14 years.
Our other guy used to snooze in the garden, and the birds were unafraid of him! DH has lots of bird feeders so we have a lot of birds in our garden which our neighbours' cats unfortunately love! They would fly off when other cats approached but weren't one bit phased by our auld fella. It was clear they never felt he was a threat, and I never saw him chase a bird once.
Some cats are hunters, and some just aren't! We had a mummy cat very many years ago, and she was a hunter, particularly if she had kittens. At her worst, she brought home 3 dead rats in a day! (This was back in olden times out in the country when cats just didn't get neutered!)
We kept 5 of her kittens over the years (don't worry, the others were rehomed) and not one of them was a hunter at all!
My mum's last cat followed her home from work one day and moved in! He never went for birds, rats or mice, but he liked to bring home rabbits. He mostly tended to bring them alive, so often they were liberated!
It's the luck of the draw I think. Most of my neighbours' cats (and there's a fair few) do go after birds and I hate it.