I always heard that called "Army" dodge ball.
That is a fun game because of everyone hopping and limping around.
With paintball not so much.
If you want to tweak the eliminations rules a bit, no problems there. I personally love playing the medic role at scenario games. It's a lot of fun running under all the fire to heal teammates. The difference, I think, is that at scenario games, there's always another objective than simply trying to eliminate everyone else. I don't think I would enjoy it in a pure elimination game.
Playing elimination only with multiple hits is a bad idea but if you want to play with flags or an objective, it is sometimes a fun twist to the game.
And no matter what, you're going to run into the issue of exactly where the arms/legs stop and the torso begins. There are better ways to mix it up than by giving people multiple hits, though.
Before: "You're playing with WHAT?"
After: "Crap! It's that guy with the pump!"