This is the weak part of your argument. You claim combat encounters are the main way to deplete resources, and therefore imply that only combat encounters matter and only combat encounters need to be attended to.
And since the solution needs to address resource attrition, combat encounters are the primary resource usage and need to be weighed much more heavily than anything else.
However, if players are leveling up by means of social encounters, then it doesnt matter if combat resources are being depleted or not. They remain unused or trivial. Relatedly, combat resources are combat resources, they dont need to be depleted if players are leveling up without them by various noncombat encounters. Combat resources become irrelevant.
Your next concern I agree with. It is the number of combat encounters that matters when addressing the class balance − at combat.
However, that doesn't address how fewer encounters affect the balance between classes.
... number of encounters affects at-will and long-rest-recovery classes differently.
"Less rounds throws off balance between classes"
There are two very different aspects that need to be met by number of encounters per day.
One of them is challenge. And yes, you can have fewer, deadlier encounters and reach your goals for this. ...
The other one is balance between the at-will classes like rogue or the EB-focused warlock, and the long-rest recovery classes like full casters plus hybrids like the barbarian or the paladin.
If you took your average full caster and took away all slots, they would be less effective on average than at-will classes like the rogue. At-will > cantrip. (This doesn't include EB boosted with invocations.)
On the other hand, if you gave casters unlimited of their highest level slots, they would do more than at-will characters. A fireball with multiple opponents, etc. Slots of the highest few levels > at-will.
Putting them together, we get, in generic terms for the average character:
Slots of the highest few levels > at-will > cantrip
So in order to balance these, we need some number of spells cast using highest level slots, and some cantrips or low-impact spells (like 1st level offensive spells in T2+). Some above and some below will average out to the same as an at-will.
...
If an encounter is 3-4 rounds and you can a spell lasting 1 minute, you only get 3-4 rounds of it at most. But if the combat lasts 9 rounds, then you are getting 2-3 times the effect from the same slot and the same action. It's more powerful. So you need to offset it with even more rounds of lower than at-will efficiency.
...
To sum up:
1. Can balance danger and challenge in fewer encounters by having tougher encounters.
2. Need to have more total rounds fighting in fewer encounters that all of the more encounters in order to maintain balance between classes.
And that second one does not often get met. Fewer encounters per day is usually fewer total rounds then if we did all of the encounters per day, and that definitely is mathematically biased in terms of the long-rest-recovery classes like casters as well as a big boost for hybrids like the barbarian and the paladin.
The above I agree with. The number of encounters and the length of each encounter matter when calculating the average combat balance between class.
Yes, the DM needs to adjudicated the level of the combat threat against the party. It is important to mix the levels of combat threats. I will discuss why below.
A trivial encounter may not count at all, while a drop-dead all-out might count for multiple.
Sure. At the same time, I feel even a trivial encounter is worth ½ encounter − even if just for the sake of how much time it takes to play the game. Meanwhile, if a near-TPK is worth 2 encounters, what is the point of awarding 3 encounters if no survives to receive it. Plus at certain point, they really do need to flee and might be foolish if they dont do it, unless perhaps there is some greater than life ethics to remain in a doomed fight.
The 13th Age solution with 4 combat encounters per full-heal-up explicitly gives the DM that judgement. (13th Age grants less daily powers for the characters at all levels, it's calibrated around 4 encounters, unlike D&D 5e).
I wish every caster in 5e was using the Warlock schedule of fewer spells at a time that refresh per rest. At that point, the Fighter too has features that are per rest. It suddenly becomes easy to balance everything.
Back to the earlier point. I agree, counting encounters is how to balance combat when comparing classes. However, whether this matters or not changes.
We are at a level that requires 15 encounters to advance to the next level. Compare the possibilities.
• 4 trivial combat encounters and 13 noncombat encounters
• 8 solid combat encounters and 7 noncombat encounters
• 15 solid combat encounters
The first possibility doesnt care about combat. These are old school "roleplayers" who enjoy the method acting, the puzzlers, the ones working on their ambitious projects, the gameworld immersion. For them, counting the encounters is the best, because they play D&D the way they enjoy it, and level up at a satisfying pace, at the same rate as combat players.
The midway possibility is the concern. It depends how the noncombat encounters intersperse. Say 4 solid combat encounters happen before the players refresh. Likely the per day casters are unloading their heavy magic and shining, while the Fighters are fresh and tough but dont get to show off their staying power. Even so, this is how 2014 is now already. Meanwhile all the players are participating in meaningful and rewarding noncombat encounters, and shining variously in noncombat ways. So even here there is an improvement.
In the total combat game, counting 15 encounters works better than ever. The math is perfect. The theoretical balance is actualizing.
In every possibility, counting encounters works better and in many cases perfectly. As I mentioned, I have been counting encounters for a while. It works well. I have seen no problems. All players have chance to shine (often by saving each other).
Even in the midway possibility, it matters when there is a combat encounter that is "too" powerful, that the players should run from. When combated, these too-tough threats tend to last many rounds before the players realize the predicament they are in. The per-day casters tend to unload their best spells. The combat resources deplete. The Fighters are starting to shine. Sometimes the players surprise, by taking out the threat. Then they really deserve the extra encounter. What happens next is the combat encounter after this. Then the casters enter it vulnerably dealing cantrips without heavy magic. The Fighters are shining. And this is happening within only three or four combat encounters.
Counting encounters helps in many ways, and is better than xp and milestones,.