I've always thought that developers are missing an obvious middle ground. Regenerating health allows a better feel for 'real injury' - if you get hit by a few bullets, you die. If you had only that much health and didn't regenerate, the game would be frustratingly difficult. In fact, there are plenty of examples of that. On the other hand, granting health pickups forces the player to conserve their life. That grants a sense of survival and long-term danger, which is also a good thing to have for immersion.
Why can't you have a stash of medical supplies, essentially a health bar or counter, that gradually heals you when you haven't been shot for a while? Basically, you make a game with regenerating health where the regeneration consumes resources. It gives the real sense of 'a few bullets will kill me', the urgency of 'low on supplies', and even a more justifiable reason that the protagonist can survive what he survives.