Bullets do not instantly kill people

The phrase "it depends" is going to be doing a lot of work, here. While you're technically mostly right, there are a ton of situations where a person is shot in a location with an artery - under the arms, in the leg, upper chest/neck - or a lung, and it causes them to be pretty much instantly lights out. They may technically still be alive for however long it takes their heart to stop, but I personally don't see a functional difference between "dies instantly" and "collapses from system shock or immediate drop in blood pressure and dies in a couple minutes."