So with the introduction of herbal remedies and alchemy, I think we no longer need food to be the primary way of healing. It's unrealistic that foods would heal wounds anyway (something that has always bothered me in video games).
So I'd like to propose a new system, in which a hunger bar is added to player status. This bar would slowly fill up over time, but would fill up more quickly during activities that decrease your stamina (running long distances, fighting, mining, etc.) When the bar becomes about half full, players start to see a slight drop in stats. As the hunger gets worse and worse, the stat penalty gets more and more severe. If the bar fills up, the player should die of starvation, and respawn with the bar like half full.
Rather than instantly healing characters, food should decrease their hunger. Basic food like apples and bread should decrease it a little, while more advanced cooked and baked items would greatly decrease it.
Potions and herbal remedies should be used for quick healing. However in order to prevent foods from losing their value, cooked and baked items should also heal gradually over time (for example, 3 hp per second for a minute).
In terms of time, hunger should not be something that a character has to worry about all the time. I'd say for a character that's just sitting down doing nothing, it should be 2-3 (in game) days before hunger sickness starts to set in, and another 2 more or so before death. Characters who are exerting themselves fully (ie running/fighting/mining continuously) should probably feel the effects of hunger after about 1 day in game.
Overall I think this system adds a lot of realism to the game, and it gives cooking/baking skills another, more realistic purpose.