This is an interesting thread.
My impression is that the result is really dependent on the type of game you'll be playing.
If your game typically includes many fights per day, several days in a row, then your house rule will gradually put more pressure on the cleric(s) to heal the party and/or will increase the gritty feel of the game.
If you want to compensate this by allowing healing potions to be bought liberally, I wonder if it's worth the change. Presumably PCs will have enough gold to buy healing potions, so it'll result in more book keeping to manage potions for a same result of PCs getting healed up at a somewhat identical pace.
If there aren't more healing potions available (i.e. you can't just buy a 12-pack at the corner store), this might result in the cleric having to use more spell slots for casting healing spells. This might be viewed as less interesting for the cleric player (or perhaps not). It might also increase the gritty feeling by increasing the potential leathality of the game, although this is even more dependent on the types of encounters than on the available healing IMO.
If, on the other hand, your game includes frequent down time, and/or few fights per day, then this house rule is likely to result in the PCs having a somewhat equivalent healing power due to more hit die to spend per day due to rapid recuperation of HDs. Presumably, if there are no 2 consecutive days with fights, they'll be back in business as if nothing had happened, HP-wise.
In my games, I typically have less dungeon crawl and often have days wherein there are no or few fights. My situation might be as per the second case scenario above. Except occasionally, when they'll end up having to fight through a castle full of opponents, and on the next day flee the scene with enemies on their backs to fight off.
Open parentheses: in my games, PCs never "sleep in the dungeon". It doesn't happen. They know they'll die if they do it. You simply don't go to sleep in an enemy's base that is still occupied, the enemy would muster their entire force and crush the PCs. So dungeon crawls are limited to a one-day affair, that might take PCs more than 12 hours mind you. Once they're out, hopefully they have what they came for. Otherwise, things get complicated, because enemies will react intelligently if their stronghold has been breached: no "let's put two guards back at the front door like before" nonsense. They are afraid that they'll DIE, as in STOP LIVING, consequently, they will try to figure out what happened, who managed to breach their lair, how they did it, give pursuit if possible and if they think it's worthwhile or if they want revenge, etc... Close parentheses.
Back to the OP's original topic, I think that the idea you're toying with is excellent, and I believe that considering HP recupration rules before starting a campaign can make for an interesting balance to keep the players on their toes by adjusting the HP recuperation with the type/number/frequency of encounters. I will certainly consider how to balance overnight healing in my campaign, with the type of campaign that I'm offering for my players. I really like the base assumption that there is no full overnight recuperation of hit points for story reasons too. And of course, as noted above, I like the added grit. However I do not wish that this results in the cleric returning to a portable font of healing, nor to potions of healing being bought to compensate the lack of automatic healing. I know my own response will probably be to adjust the encounters themselves once the campaign starts, since I'll be homebrewing anyway (even if I use a module as a base storyline).