You must take a wizard relative to a fighter in a 6 encounter day and compare that with a wizard relative to a fighter in a single encounter day. You seem to be saying a wizard does more per round in a single encounter day than a wizard in a 6 encounter day - which while true isn't a useful point by itself.
Thank you for your measured and reasonable tone, I will try to match it. I think we got a bit adversarial.
That's not quite the points I was making. Let me try to be clear.
1. Characters take more Actions during encounters over the course of a 6-8 encounter day than during a 1-3 encounter day. If in both cases spell slots are expended by the end of the day, that means in the 6-8 encounter day more Actions were filled with Cantrips, which (baring class feature boosts like the warlock) have less effect than on-level and near-level spells.
Averaging in a lesser effect means that the average effect per Action is less for a caster during a long day.
Casters having a variable effect per day means that their balance relative to primary at-will classes change based on the number of encounters per day.
2. Spell slots and the actions to cast them
can be (not
are) more efficient in larger encounters.
2a. A long lasting spell that lasts for 8 rounds will have more total effect than one that lasts for 4. That's more likely in harder encounters as they tend to run longer.
2b. If encounter deadliness is increased via more foes, area of effect can do more total damage/affect more foes per casting, while at-will actions are single target.
2c. If encounter deadliness is increased via more powerful foes, due to the nature of creatures in 5e having 2-3 bad saves the proper spell selection can make more powerful foes just as easy to effect as weaker foes. So the same spell slot can action deny/debuff/or otherwise impair a more powerful creature. At-wills usually target HPs via AC, both of which usually increase with a more powerful creature so they don't have an outsized effect.
Taken together, this means that starting from the baseline of a spell slot having the same effect regardless of encounter deadliness, we can see that there are cases when a slot can be even more effective. This can, when it takes place, further increase the effect per Action for casters in ways that the primary at-wills usually do not match. (They can - for example a grappler build can go after a more powerful but physically weak/clumsy foe. But that's not "business as usual" for most groups.)
One great point you brought up I didn't address is short rests. I've seen them all over the place from after every encounter to once in about 12 encounters (just two weeks ago - we were against the clock).
Instead you should be considering the impact the wizard is having on the single encounter day and compare that with the impact the wizard is having on the 6 encounter day. Impact is not measured by looking at their actions alone but also at what heir allies and their opponents are able to do. So the question is: how do we measure impact. So far you've disagreed with any method I've suggested - but while not perfect I'm finding my measurement methods for impact far superior to your ideas around it.
I think that's really where we need to begin. How should we be measuring impact.
As a simple example: consider the impact of doing 10 DPR against an enemy with 100 hp. Now consider the relative impact of doing 20 DPR against an enemy with 200 hp. I would say both characters have the same impact on the fight at hand. However, throw in an ally that does 500 DPR in the first fight and one that does 20 DPR in the second fight. I would now say the first Character has less impact than the 2nd.
So then impact IMO is a function related to your own actions, ally actions and enemy actions. Any measure that isn't accounting or comparing for all 3 will be fatally flawed.
So how do you propose we measure impact?
When modelling is hard because of all of the factors, we can go to original source. Play a few high encounter days with your normal players. Do 6-8 encounters between long rests with two short rests about 1/3 and 2/3 of the way through. Tell them ahead of time so they aren't blindsided as that would put a spin on it. I mention a few, because the first might have lessons learned that will impact how they play in future ones. See if there is more cantrip usage. See how the at-will characters feel.
(Note: Some things that may impact this. If you are playing Tier 4 or higher Tier 3, spell slots may be so plentiful that cantrips aren't used as much. Also, check if your players actually have a good mix of primary at-will and primary long-rest. In one group I play in the DM does 1-2 encounters in a regular day, and most of the player tend to play something that can nova in order to survive the high deadliness, so that group doesn't have a mix.)