I don't. I think it's inescapable math.
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. Simply, an At-will > cantrip. (This doesn't include EB boosted with invocations, that's a cantrip plus class features, just as martials have class features that boost their weapon attacks.)
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.
No one debates that. And it has nothing do to with campaign style.
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 plus 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.
Let's examine that. If you run a few encounters and run the party's casters all the way out of spells - you are STILL not balancing the classes unless you also are forcing them to have a good number of rounds casting cantrips - it needs that "less than at-will effectiveness" to balance out.
An easy way to work this out is average effectiveness per action, over the course of the adventuring day.
Ah, so if you have fewer encounters, as long as the last as long as more encounters we're good, right?
Well, no. It's moving in the right direction, but durations are a thing. If an encounter is 3-4 rounds and you can a spell lasting 1 minute, you only get 3-4 rounds of effect from 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 than if you were just doing more encounters.
A easy way to see this is the barbarian. Say you've got 3 rages per day. Assuming the encounters total to the same deadliness, is there any case where you are worse off if you can rage for every encounter instead of half of them? That's one of the things that decreasing the number of encounters does - allows duration effects to be even more powerful.
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.
I disagree with your analysis on several levels.
First, I don't think "average effectiveness per action" is the one and only barometer of intraparty balance. I'm sure it's at least a large consideration for some players at some tables playing campaigns run in certain styles, and I suspect it's likely the
primary consideration for a lot of people. But focusing only on that parameter ignores other barometers of balance, such as "which characters' actions were outcome-determinitive" or, at a more meta level, "which players' contributions were outcome-determinitive" either or both of which can impact perceptions of balance for some players more than average effectiveness per action.
For instance, in a fight where the outcome is in doubt, (in contrast to the usual expectation that the PCs will handily win), it's entirely possible that, without the contributions of
every PC, the fight would be lost. Their "average effectiveness" on a per-action basis might be wildly different (particularly if any of them were disabled or forced on the defensive for some or all of the combat) but each were equally 100% crucial to the party achieving their objectives. Maybe that doesn't matter to some players' sense of balance, but it will to others, and campaign style will have a heavy impact on how often it is that every character's actions are simultaneously outcome-determinitive.
Second, you're leaving out any consideration of contributions to success in combat that happen before initiative is rolled. I discussed in my previous post how collective battle planning impacts balance, but your analysis doesn't take it into account. As an example, if the player of the rogue comes up with a plan which allows the party to steamroll without risk an otherwise-challenging combat, but that plan involves the rogue distracting a critical NPC to prevent them from joining the fight, the rogue's "average effectiveness per action" is zero (or, at least, exceptionally difficult to quantify) despite the rogue arguably being single-handedly responsible for the victory. (Other examples include encounter-defining spells, such as the Ranger's
Pass Without Trace turning an otherwise impossible fight into an easy ambush.) And I'd emphasize that campaign style
heavily influences the extent to which planning and actions taken before combat are allowed to impact the difficulty of the combat itself.
Third, even when one does use "average effectiveness per action" as the barometer of balance, the rest of your analysis assumes that battle conditions permit unencumbered use of the party's highest-level spells. If a caster regularly has to decide between sub-optimal use of one of their highest level slots versus spending multiple actions to set up an optimal use, that will directly impact your calculations of the average, but you haven't taken that into account. In other words, you don't necessarily have to run a caster out of their best spell slots before their effectiveness per action drops below that of casting one of their best spells, which in turn necessarily lowers their average effectiveness below what your analysis expects.
The simplest example of what I'm talking about about is whether and how often opponents use full cover, which severely impacts the usefulness of multi-target spells to a greater extent than it does attacks (since the attacking characters can switch targets and only lose the effectiveness boost from focusing fire). For instance,
Mass Suggestion is famously capable of ending entire combat encounters, but if it takes multiple actions and/or spell slots to get to a position where you can see and target all the enemies, its "average effectiveness per action" drops sharply as the denominator increases (or else the numerator drops sharply if you use it on only the subset of the enemies immediately in range and visible). How often creatures use full cover (or other tactics that go beyond a short-range brawl) is again heavily dependent on campaign style, which thus influences balance even under your preferred barometer of "average effectiveness per action".