I'm at a friend's place and the cat keeps bringing in dead (or half-dead) animals into the house. It's my understanding that cats think we are big, helpless kittens that don't know how to hunt. Hence, they think they are doing us a favour.
It seems like a few mice actually escaped and found refuge in some walls in the house, so these "presents" are actually more than just annoying (and smelly if the dead animal ends up behind the couch).
Cats outside are an aggressive invasive species in every continent except Africa. I love them but they need to be kept inside, and neutered/spayed. Cats do fine indoors, even ones who have gotten used to going in and out. They may complain but with enrichment and regular play they will get over it. If not, then a catio is a good solution.