Shooting isn't the core mechanic in the majority of video games.

I’m curious if you understand what majority means?

Exactly this. Anyone who thinks that must have almost no real exposure to video games. The diversity of gaming experiences is huge, and shooters represent only a small fraction of that.

Indeed, as we all know the order is

1. Run

2. Think

3. Shoot

4. Live

The most prolific violent video game is Minesweeper. The fidelity may be low but the little guy dies and mines are powerful weapons detonating all over in the game.