I think a way to justify "Healing by Killing", especially for a fast-paced action game, is to have health represent some sort of "Strength/Resolve/Endurance through Self-Confidence and Spirit," since it's already established that no health system can be realistically implemented and fun at the same time.
It's also the most sensible from the abstract when the point of the game is kicking names and taking ass, because since losing health is punishing screwing up and failure, success and doing things right rewards you with more health.
I like an idea you didn't mention, but I thought of when reflecting on Halo's system: Give two health bars, either a "Health from Time/action" like the Walk it Off, Health-through-Murder, or Regeneration systems, and one "Recover-through-depletion", like immobile health packs, med-stations, and usable Healing Potions; or a "Easy, short-term recovery" and "Hard, long-term Recovery" system, like that seen in Left 4 Dead, with its pills and first-aid kits.
Even my favorite tabletop game, D&D, uses a Dual-health-bar system, with Standard Hit Points and Temporary Hit Points on top of those.
And: Halo 1 had a better system than Halo-Reach, by not extending the Shield bar to also include half the health bar.