In contrast, gamers of my acquaintance don't have that expectation, so don't play as if they can rest any old time. Instead of "going nova", PC's carefully manage their resources, always maintaining a reserve. For example, going through the 3.5 version of the ToEE, it wasn't uncommon for our party's mage to still have spells to cast after 5+ encounters. Which was a good thing because we (the Party) didn't determine when we got to rest- the campaign environment did...and there were times when we were retreating carrying one or another comrade from the field of battle. Sometimes harried as we went.
With this kind of playstyle, the proposed change reaaaaaally doesn't seem necessary. In fact, it is kind of anticlimactic.
Yup.
Like Justin Alexander says in Death of the Wandering Monster there are only two ways to prevent the 15 minute adventuring day:
(1) Eliminate any advantage from the "1 encounter, then rest" model of play.
(2) Apply meaningful consequences to the decision to rest. (Wandering monster encounters; NPCs trying to hunt them down; or simply NPCs reinforcing their positions, closing up shop, destroying evidence, etc.)
The latter is in the hands of the GM. The former could be solved mechanically, but only at the cost of sacrificing mechanical strategy.
Another way of looking at the problem is this: There are two varieties of the 15-minute adventuring day. The first is the "nova" (the PCs use all of their most powerful abilities in a single encounter and then rest); the second is the "forced rest" (the PCs are out of resources and must rest/retreat to recover them).
These are two different problems.
Fixing the Nova: Mechanically you can fix this by removing all daily powers. Non-mechanically, the nova becomes a valid strategic choice (allowing the PCs to defeat more powerful foes) that isn't always the best strategic choice (if it allows NPCs to regroup, escape, or destroy evidence).
Fixing the Forced Rest: Mechanically you need to remove the hard limits in your system design. At the very least, this means that hit points need to completely regenerate between encounters.
So what would be the absolute worst way to fix the 15-minute adventuring day problem?
(1) Give all classes daily powers (increasing the mechanical incentive for nova strategies).
(2) Create a hard cap on the amount of healing any single character can receive (introducing a hard limit resulting in forced rests that didn't previously exist).