Oh come on, when did you try to drink somethink from a bottle last time? Not a plastic one, which will shrink as you drink. You know why it does? Because it happens when you don't let the air in. Now, try drink this way from a glass bottle

after the first sip, shot, or whetever, hehe, the lower preasure will make it harder for you to make the next one. You need to let the air in and it takes time. I posted the small bottle of votka to visualise you the process of drinking. Just think how you drink a beer from a small bottle. Try to drink it as fast as possible and you will know how you would drink potions.
Besides, first you need to reach it and as pointed out you need free hand.
Another thing is how it tastes like. It is purely alchemical recepture (So bilbous, before you go with your instant magic effects of uber wonders, please be so kind and step away with it from alchemy, as there is a lot from chemistry and biology in it, what you know aren't magical forces i our universe. You have completly no arguments why magic should be instant, besides that you want it this way and you have such feeling

).
The element which speed up the process of assimilation is alcohol, so i won't be suprised of it wasn't part of the potion. Now think about the votka again, if you ever drank it, you know that it isn't good to drink a lot of it at once.
Shields. I have been fighting using a shield and i can assure anyone that holding it, you can do nothing else with that hand.
EDIT: There is also matter of what the HP really are.
While for me it is some kind of mix of level-of-being-tired/being-able-to-see-if-you-are-loosing-or-winning/being-trained-in-keeping-the focus-while-being-tired.
For others it is health. Extremaly simple explanation for people who are lost, when the logic is involved. Health which you can loose only after being badly hit, what basically means end of the fight. This makes wonder why the bar is ever necessary, coz you always loose it whole or none.
But that is talk for another discussion. A discussion like this one, which has many copies on these boards.