No.
Medkit or no medkit, if the game is made right, I care not.
Well for me personally I feel games like Bad Company 2 are made right, but the health system makes them more boring, that's all.
online wise or single player wise?
Both?
At least in the case of Infamous or prototype,its there but there are better alternatives since its so slow(like draining electric or consuming people)
I really actually enjoyed Prototypes health system. But then again I'm not really talking about that genre.
My theory is that your health regenerates, but does so at an extremely slow speed
Except that just makes people wait for longer.
FPS's like Team Fortress 2 and Unreal Tournament 3 don't have regenerating health.
Gr8 gr8ps
Personally I really enjoyed Far Cry 2/MoH: Airborne's stance on it. Your health is divided up into sections, and once one is depleted it no longer regenerates.