It helps a bit if those fewer encounters are also longer, since with more combat rounds the resource expenditure generally goes up. (Although it does depend on why the combats are longer--a running fight when everyone has full cover most rounds is going to take longer without necessarily using up more resources.)
Sure, a touch, but not as much as you'd think. Especially since even a longer encounter will rarely exceed 1 minute (10 rounds) which is the length of the shortest non-instant spell durations.
If a barbarian has 3 rages a day, he can rage half the encounters during a 6 encounter day. Cut that to three encounters and she's raging for every encounter in a day.
Casters puts a particular buff and party member(s) every encounter. Takes 3 slots for a 3 encounter day, 6 slots for a 6 encounter day. IF they have enough slots of that level or higher, it means that at least 2 were upcast. Many buffs don't even benefit from upcasting.
If the encounter is more difficult because there are more enemies, an area of effect spell can catch more of them. Even with saves, that's a lot more damage done per spell slot.
If the encounter is more difficult because there are more powerful enemies, it's even worse. The nature of only 2-3 good saves for monsters (between proficiency and having high ability scores in non-proficient saves) means that there are always weak saves. With spell DCs increasing with proficiency and weak saves not, you can affect a powerful as easy as a weak foe as long as you pick the right spell. Casting 101. So, for the same debuff/crowd control spell you are able to stop a more power foe (or foes) - again, getting more result from the same slot.
Ongoing spells that are target-able like Spiritual Weapon and Call Lightning also don't require additional slot resources to do more damage for the harder battle.
This isn't saying that more resources aren't used in a longer fight. They are. Three long fights will use more resources than three short fights. But six short 3-round fights will use a lot more resources than three long 6-round fights.
What you said is a common attitude that unfortunately doesn't hold true in actual play. And so few people have experience with regular 6-8 encounter days that the myth persists. That is the point
@Don Durito made - actually having a "standard" 6-8 encounter day turned his fighter from constantly overshadowed by casters to the MVP.
If not convinced, try it for yourself. Have several days of 6-8 encounters, with two short rests about 1/3 and 2/3 of the way through. Experience how it differs from a few, long fights.