The only games I've played extensively, with regenerating health, were Call of Duty 4 and Gears of War. In both those games, the speed of regeneration made everything a lot more boring and challengeless than it needed. While CoD4 kind of maintained the fun due to the different set pieces it threw at the player, Gears of War was made even more repetitive and dull by the regeneration, than it would be without it.
Where I'm going with this is that health regeneration being good or bad is, probably, simply a matter of implementation. Also, bad health regen doesn't necessary break a game. All of that said, I have yet to play a game where I felt health regen was well implemented.
Edit: Also, I wouldn't call GoW and CoD4 outstanding proof that medkits are a thing of the past. If anything, they're proof that regeneration needs to be fundamentally changed if it's going to work.