I believe that regenerating health has its place. Let's look at why it was invented.
In the original Halo (and its a great game), they provided energy shields to fix the problem of a game becoming unwinnable. In most previous shooters, you had a limited amount of health, and while there might be medkits and such, they were also finite. You could wind up with a single hitpoint and have no way out of a situation, no matter your skill, other than restarting the entire level/game. And in a fast-paced actiony shooter, that's not exactly fair.
Wh.. why not? It seemed to work fine in Wolf3D, Doom, Quake, and any number of other FPS games prior to the implementation of regenerating health. In fact, I would argue it even adds an additional level of tactics in multiplayer: if you fail to kill your target, you expect that he's going to go find more health or armor instead just hiding in a corner somewhere. If you're familiar with the level, you can anticipate where he might pop up in order to catch him.
How is it not fair? Really, most of the games I've played with medpacks really inundate you with them. Even on hard difficulties. If you get stuck in a place that you can't advance because you lost too much life... I guess you'll have to... play better? Why is demanding the player to restart the level so odious as to be avoided? I can understand from a certain point that it may break immersion, but in that respect regenerating health (and in fact any kind of health recovery) can break immersion. I can understand slightly better if there's an unavoidable environmental hazard, but then that's just bad level design if the environmental hazard is unavoidable (otherwise it's just a... penalty, in the form of level design, not necessarily environmental nor a hazard, which each imply avoidability).
Taken slightly further, the concept of health restoration (whether via kits or regeneration) is only an abstraction to give the player some leeway in making mistakes. In fact, the regeneration mechanic can make sense if you take a slightly different tack: rather than define as damage the player takes, from which the player recovers, define it instead as the player's "panic" level: it's not so much that he just took a bullet to the head, it's that the bullet came
really close to taking out his head, and that freaked you out something fierce. In that respect, taking a few seconds to breathe and calm down before popping out again can make better sense than actually regenerating health, and even implications for becoming more stoic in the face of a bulletstorm. Of course, such an interpretation requires refinement to function well in a wargame.