The rate of health regeneration could be tied to the difficulty level being played. The level of regeneration would be high on the Easy setting and none for the most difficult setting. It could also be designed to be set completely independently of the overall game difficulty.
Unfortunately, it's not that simple. When you design a game around regenerating health, it's probably going to be pretty broken when you remove regen entirely. Unless you replace the regen with some alternate replenishment system, some levels would likely be impossible.
That aside, I agree that regenerating health makes games too easy. In any game with regen, I find that I don't care about getting hit unless I'm about to die. There's otherwise no long-term consequence to getting hit. It's boring.