The problem with removing homogeneity is that there are also other goals that the system wants to achieve. The system aims at a balance between characters. And this balance is seen "short-term" - e.g. per encounter as the smallest relevant unit. I think going away from this goal is not desirable. You are just removing one issue and replace it with another.
One take to deal with the homogeneous is to change the rules for resting.
For example, let's say in the current setup a at-will has the power of X, an encounter power has the power of 2X, and a daily power of 3X.
A typical combat might be 8 rounds. Character A has 3 encounter powers and one daily power. Character B has 1 encounter power and 2 daily powers.
Character A has a total power of 4X +6X +3X (=13X) in one encounter, and 4X + 6X(10X) in every other. Character B has 4X + 2X + 9X (15X) in one encounter and 4X + 2X (6X) in every other that day.
If the characters go through 3 encounters before taking an extended rest, Character A is better of (power = 33X). If the characters go through 1 encounter and then take an extended rest, Character B is better off (27X).
So, it is in interest of Character B if they take an extended rest early. It is not exactly a bad idea for Character A to do it, too - he has, after all, a daily resource. (And that is true with 3E hit points or 4E healing surges). But in gameplay, it turns out Character B is more effective then Character A.
If we remove the concept of extended rest/sleeping as the way to recover long-term resources, we could get away from that. Say hello to Milestones.
Every time a character completes a Milestone, he could get back his "daily" powers. If we use the 4E definition of that, this means every two encounters.
Character A could be a Fighter. Being a martial character, he of course has no daily powers, that would be ridiculous and unrealistic, as everyone knows.

He has 4 encounter powers (each 2X power). Between Milestones, this is 8X.
Character B is a Mage. Magic is very difficult. In combat, he usually swings a staff. Magic is difficult to control, and it takes a while for the mind to recover from casting a powerful spell. So he has 2 daily powers, each at 4X power. Between milestones, that's 8X.
Character C is a Cleric. Divine magic is not as strong as arcane magic, but the gods provide simpler prayers that can be done in a few minutes by a true believers. C has 1 daily power at 4X and 2 encounter powers at 2X. Between Milestones, that 8X.
Of course, now multiclassing is probably a mess.

And does it make sense to have Healing Surges recover too at this pace?