I agree with your analysis of the Fighter's role. Now here's the question: can other classes do it better than the fighter?
On the short term yes, on the long term...no.
This is why in 4e there was a push to do away with the one encounter day. Even two or three sometimes is too short.
In the movies...Do you see Conan go to the dungeon and after 6 minutes call it quits?
Sometimes adventurers are in the dungeon or adventure for days. Also, enemies aren't stupid, though many play them that way. Goblin guards killed at an entrance don't just leave an entrance unguarded after that forever after...somone discovers what happened, more are sent to guard, and others are sent out for vengeance. You don't get that chance to recover...
That's where the Warriors come in.
Plus, anyone who doesn't want to play as a group or in a group and wants to play a solo spellcaster can go play with themselves in the corner all they want...but most want someone who can play with everyone else.
Everyone complains about casters at the upper levels, but the thing is, casters have to GET UP TO THOSE LEVELS first.
The dynamic would be more like Fighters are great at the lower levels. Most casters will die in relation to the fighters at low level. At high level, if I am a Wizard, I want to have that power to be able to crumble an entire city at my whim...but then by that point, I'm high level, and yes, I would want my power to overshadow that of a normal fighter.
If one wants to destroy the Wizard or Cleric or Druid (but why a Druid...they should't be inherently good OR evil) then there should be a quest that requires something extreme...perhaps to give that fighter the special item that allows them to fight the spellcaster on equal ground.
Heroes in books normally have to find that special item or they'll be toast as well. I like that dynamic.
It's sort of a reward for those who played that 1st level Wizard that would get squashed after their two spells were cast if it weren't for the Fighter to protect them, or the Cleric that had to choose which spell to cast, one to aid in healing or one to aid in battle. They get that all power at the end of the powerups...after playing the weaker guy at first.
However, it should be remembered there's ALWAYS that equalizing dynamic in the adventures. Perhaps it's that one Ring some halfling will toss into the fire. Sure you could toast Aragorn with a word...but that halfling will catch you by surprise. Or maybe it's that magical Sword conan finds which makes him completely immune to any magic you cast. Sure you can annihilate his companion, but you better have someone to protect you against Conan or you'll simply be another one of the many deaths to his name.
Okay...I got side tracked. Yes, I do like the dynamics of spellcasters getting more powerful...BUT I think Fighters have their place and purpose and even that Cleric or another can't really replace them. I've seen people try, but most of those get banned (actually all of them have been banned from every group I've ever seen) for being really selfish and not good team players.
And that's the REAL importance...is that you should be a good team player and play as a group. Every character has different strengths and abilities. An RPG shouldn't be purely about combat, and a spellcaster that concentrates solely on that is wasting a ton of potential. I'd rather have a Fighter that can kill the things, and a spellcaster that can save us from being killed from that infamous puzzle by using the right spells at the right time then some cleric that thinks they are a better fighter than a fighter any day of the week.