Off the top of my head...
We already will have to learn how to introduce a somewhat arbitrary "time unit", the ENCOUNTER. (And I LIKE that.)
How about introducing another arbitrary (as in "not fixed, as long as narrative logic requires), longer unit called the ADVENTURE? Not necessarily meaning the same thing as Adventure means in "published adventure"; for example, if a published adventure includes investigating the sewers looking for the thieves guild, following the trail deduced by the notes found there to the bandit king castle, and storming the castle, each of these three "items" could be an "Adventure".
Once we introduce the concept, we might simply say that DM keeps track of the total number of healing surges a character uses during the "Adventure"; once the Adventure is over, the adrenaline that's kept the PC standing and fighting and pushin onward is REALLY exhausted. He (she) needs rest for a day for each healing surge used (or something like that). Healing surges activated by magical healing do not count for this calculation.
This seems to me not to add any unnecessary bookkeeping (counting healing surges is as simple as putting a small cross on the character sheet every time one is used) and does model, a little, the "we need to rest and recuperate" which I think CAN HAVE a place in the game.
Modifications might be made to the rate (don't know, maybe one rest day per 2 healing surges, for faster recuperating), or to the factors (if you've been hit by any number of criticals, that alone requires, say , a week's rest).
Just making suggestions, of course.