8 encounter a day is how WOTC keeps wizards from outdoing fighters in combat.
even at 8 encounters a day wizards still spend more then half there rounds using 3rd+ level spells...
8 encounters 2 short rest so lets say 3 encounters rest, 3 encounters rest 2 encounters call it a day.
at 20th level the wizard has 5 scaled cantrips (so either 4d10, or a lower die code with an effect) and 4 1st level spells (plus 1 turned into a cantrip/at will) 3 2nd level (plus 1 turned into a cantrip/at will) 3 3rd level spells (plus two that you can do 1/day each for free) 3 4th level spells 3 5th level spells, 2 sixth level spells 2 7th level spells 1 8th level spell and 1 wish... I mean 9th level spell sorry don't know how I made that mistake.
you then have 10 levels you can recall of spells that are level 5 or lower (so 2 5th, or a 4th and 2 3rd, or any other combo)
if we assume you are not novaing out and just speed blitzing your highest spell every encounter, and encounters average 4 rounds of combat here is what that looks like (remember I said average 4 maybe 1 goes to 6 but one ends before your second turn) and these are ofcourse rounds you CAN take actions so again average, cause maybe you spend a round stunned or something this is 20th level after all..
now in another thread I just did fireball (not the most optimal choice of spells) and said you can throw 20 of them... but lets not count WHAT spell I mean sometimes you will have optimal choices sometimes you will have bad choices and sometimes you will just make a mistake...
right off the bat you have 2/2/1/1 so of those 8 encounters 6 of them you can use a 6th level or higher spell if you limit yourself to 1 per encounter. I will spread those out and say you use 1 every encounter for the first 3, after your first rest you slow down and spend only 2 on your next 3 and the final one in one of the last 2 encounters... in each group of encounters you have a big 'shenagins' spell... but not every encounter.
for level 5 and lower spells I will say you use your arcane recover to recover 1 4th and 2 3rd (not unreasonable) so out of first 3 encounters I will use those... 1 4th and 2 3rd
so we start off at 12 rounds of me acting over 3 encounters in the morning... I use 3 high level spells... I use 1 5th 1 4th and 4 3rd (remember 2 of those are free) that leaves me 3 actions to either defualt to cantrips or other at wills or use a 2nd or 1st level spell..
after lunch hour I am still going strong and I have 12 more rounds of me acting in combat over 3 encounters.... I use 2 high level spells I use 1 5th level spell and I can reuse my 1 4th and 2 3rd (since I arcane recovered them) so I will add 1 more 4th level spell in and have 5 rounds left to use at wills or 1st and 2nd level spells
so last two encounters after dinner break and I have my 8 rounds 2 high level spells and 1 5th level spell left and for those last 5 rounds of actions I have 1 4th level spell, then I am for those last 4 rounds stuck with low level spells..
now low level spells can still be hold person (2nd) that can be at will that can be devistating. Heck Tasha's Laugh can knock out a monster with a 1st level spell that fails a save and saves are not spell level based anymore. that is before we count crazy things like mage armor lasting all day and shield can become atwill (so for the cost of 1 1st level slot and your reaction you can have 10+dex bonus to your AC in a game where a 20 AC is awesome).
This also doesn't count the times that useing normal cantrips is just a GREAT option... against a hoard of smaller threats blasting for 4dX (maybe with an effect) is pretty awesome and useing a 3rd+ level spell could be over kill... now you will target 1 monster and the fighter will on average do the same damage in 2 hits and have 4 attacks so can take down 2 to your 1 so hey... that is a cool moment for the fighter while you hold back and let him handle your light work...
that also assumes we are just worring about combat. What about a day where those 8 encounters are 2 exploration, 1 social, and 5 combat... I mean yeah those 5 combats the warrior has more HP and AC and hits harder per hit (unless it is a high level spell round) but in the 2 exploration and 1 social encounter you have auto win buttons he doesn't (yes if you know/prep them) and if you choose not to use them you both have skills, and RP to fall back on.
The day that you have to do the reverse though is a BAD day for fighter/wizard comparisons... have 3 social encounters and 2 exploration encounters and only 3 combats and that fighter only has 3 fights and the wizard CAN drop some or all of there high shenagins spells on them...
A travel day (as early as 9th level) can already be a blah day for a fighter... 1-2 random encounters plus some exploration for a class that needs a drawn out set of encounters and has little to nothing the others don't have for exploration...
edit: I just remembered it's 6-8 encounters... man, the 6 encounter day where even 1 is not combat is going to REALLY feel lopsided.