I dont know what games you are playing that even allow you to "constantly run away and hide", but I havent played any of them.
I see. So you haven't played:
1) Crysis.
2) Halo 1 & 2.
3) Call of Juarez: Bound in Blood.
4) Call of Duty 2,4,5.
5) Wolfenstein.
it sucked ass when you got to a particularly tough area when you are low on health and keep dying because some cheating AI keeps picking you off before you can find the next health pack.
The problem there isn't with finite health, it's with the cheating AI.
Crawling through a game with the cycle of die, reload, die, reload, die, reload, die, reload while trying to find some health is not fun.
That's why you shouldn't die. Finite health encourages you to play better. Improve your accuracy, reduce the amount of hits you take. When you have 10 HP and have to clear out a room of enemies, you have no choice but to play better. Conversely, if you have regenerating health, this is never an issue.
If you don't give the player a place to run away to, then that problem is solved. Most games solve this problem easily.
Again, I'm not sure what games you've been playing. The majority of the ones that I've played have plenty of cover and places to run away and hide. It only takes a few seconds to completely regenerate your health so any cover will do.
Running and hiding is not always the answer, nor is it always available.
Not always, but most of the time. Taking cover for a few seconds and completely regenerating health is an option the vast majority of the time.