Have you ever seen a domestic cat kill birds? They kill them for the excitement. And you can bet the cat does not like to give the pigeon a chance, and they mostly don't eat what they catch.
If you are going to anthropomorphize animals, you are well on the way to conceding that they have equal rights to huumans, which was probably not your intention.
Cats are cute and furry when it suits them, but still basically highly evolved killing machines. There is little difference between a wild tiger and a domestic cat apart from size.
If a cat sees something catchable they will attack it, they're not going to think to themselves "ah, sod it, I know I've got some cat biscuits inside so I can't be bothered catching.that plump pigeon".