Halo actually does have medkits. I'm betting the recharging health mechanic dates back to when it was a PC/Mac title anyway.

Yes I never played Halo but I was sure it started this. Weird. This seems an interesting topic to research; who did first regenerative health and why? By the way, does that apply to Halo 2, Halo 3 or Halo:Reach?
Personally I don't like modern shooters, too many gimmicky game play elements that aren't fun at all. Cover based shooting, vehicle sections (can be good if done well, but really means
well) regenerative health, aiming assist and sprint limits. Weapon limits are OK if they make sense in the term of the game, i.e. I don't complain about them in Ghost Recon. I have ranted formerly about the stupidity of grenade based stuff on "Realistic" shooters, I'm yet to see a grenade throw into a building from the front door or from the window without needing to show myself at the door hole.
For me, Painkiller seems to be the best thing that has came out for a long time - game play wise or level design wise. I haven't played Unreal Tournaments though, but from what I have seen, I'm not that much into RPG wars.