I like the health regen as a convenience in the game as long as it only regens AFTER combat. That will be how it is in my skyrim mod if i can do that. no regen of heath in combat but fast regen out of combat to prevent the boring gulping of potions before moving on to the next room. I have nothing against potions, in fact alchemy is my primary skill in Skyrim after blocking!
And just like I did in my Oblivion mod I will force some kind of drinking animation. At first when I did this in my Oblivion mod I hated it, but wow did it change the way I approached combat.
It adds realism to the magic that helps to make the magic more magical...if you know what I mean.
I think Psymon's suggestions are really good. As a side comment but in Demon's Souls(Prequel to Dark Souls) you could mass obtain healing items, play really careless then heal up en masse.
That game also had a "drink" animation" but it was shorter than what they have in DkS. So the balances they implemented to discourage "Careless play followed by potion chugging" were:
A) Drinking animation is slower, making it crucial to find a "safe spot" to heal. 90% of the time, if you healed at a bad time you would be punished for it.

Drinking heal animation heals late in the animation, as opposed to early on, that way you might get hit as you pull out the potion if you did so at the wrong combat time. Defeating the purpose of the potion.
C) Not 100% sure, but I believe you would take some extra damage if hit while healing.
D) Can only hold a limited number of healing items at a time. Refilling healing items respawned all enemies.
E) Healing was more powerful though as a balance, and your healing items could be stronger with upgrades.
F) Magic could heal you, however healing magic had even more cast time than potions! But again, they were powerful as a balance.
It's not possible to really prevent all kinds of "item hoarding" to cause limited potion holding, but a suggestion I have would be to make potions toxic if too many are used in a certain timeframe.
Maybe something like...if there are too many potions within the bloodstream, the results are "toxic" due to how they affect the body.
The limitation could be either...how much hp was healed within a certain time frame, or alternatively how many potions you've chugged in the last...in game hour or so. If the person chugs over
that particular amount, the potion will damage instead of heal them, or perhaps not heal them at all.
What do you think?