That's not even remotely what I'm saying.
Look at it from this perspective: the "wizard' class is profoundly broad, being able to do pretty much anything he wants with the possible exception of healing (depending on edition and supplements used). But does it suck? No. And it shouldn't. And even though the wizard is the poster child of "casting spells" he's not the only one. Some concepts do get put into other, more specialized classes.
The "fighter" class ought to be in a similar position. Fairly broad, some more specialized concepts put into separate classes, and most importantly: shouldn't suck.
The idea that a generalized class is more likely to suck from an effectiveness point of view is completely false. And note that I'm not even talking about "everything we don't have fun rules for", but instead just the core concept of a martial warrior.
All this depends heavily on how the systems are implemented (compare the classes from D&D with those from Alternity or Anima, for example). Your statement assumes far too much.
While I'd happily agree with the idea that the fighter shouldn't suck (and would hope that it's not a terribly controversial position to take), to describe both the Wizard class and the Fighter class as broad is to conflate two very different meanings of the term.
The Fighter is, traditionally, conceptually broad. It's one class that's expected to cover a wide array of archetypes: soldiers, non-mystical knights, thugs, non-mystical archers, swashbucklers, gladiators, samurai, town guards, farmboys, mercenaries, pirates, officers ...
However, it's mechanically narrow. It provides tools for being good at hitting things with weapons, and for being good at surviving being attacked with weapons in return. Fighters are good at fighting fighters. But there's generally little or nothing to support the non-combat aspects of any of the archetypes that are folded into it, and even its combat abilities are very focussed specifically on weapon-use, rather than tactical acumen, combat mobility, and limiting enemy options.
The Wizard is mechanically broad. Want to do something? There's a spell for that. There are spells for dealing damage, reducing damage you take, avoiding taking damage, stopping enemies while bypassing the AC and HP mechanics the fighter is restricted to dealing with, reshaping the battlefield, overcoming obstacles, finding things, hiding things, identifying things, making things, destroying things, repairing things, altering the colour of things, making people like you, making people fear you, turning invisible ... pretty much any effect achievable within the system, other than magical healing, can be achieved via a wizard spell.
However, it's conceptually narrow. There's no pretence that the class covers, or has ever covered, every concept of magic users that we don't have a more specific class for. It doesn't even cover the entirety of the concept of wizard. Ask a non-D&D player at random to name a wizard. Chances are you'll get one of the following names in return: Merlin, Gandalf, Dumbledore, or Harry Potter. None of those conform to the D&D interpretation of the concept. Oh sure, Merlin, Gandalf, and Dumbledore might have the high intelligence that's expected of a D&D wizard, and Dumbledore is definitely a scholarly type, but Harry's about average at best in the intelligence department while still being a more effective wizard than his more intellectual peers. The elder three of the set are also generally known for their roles as wise advisors, while high Wisdom is not a trait commonly associated with D&D wizardry. And of course, they all use radically different magical systems that aren't even the slightest bit Vancian. Unlike the D&D wizard, for whom magic is something external learned through study, all of them (well, Merlin varies, depending on which of the numerous interpretations you go with) are themselves inherently magical beings, more like a D&D sorcerer.
Now, of course, expecting a class that originates in the 70s to model characters not created until the 90s is silly. And those who read deeper into Tolkien, as I expect many of us here have, find that Gandalf is closer in concept to an angel. But to the general public, they all readily fit into the concept of 'Wizard', never mind the even broader concept of 'Magic-User', at least as readily as all of those various martial combatants fall into the concept 'Fighter'. There's countless other (less widely recognized) examples one could draw upon, both from before and after the creation of the D&D Wizard. I could also list dozens upon dozens of characters and even entire categories of characters that people would be quite content to apply the label 'Magic-User' to, and aside from the writings of Jack Vance and licensed D&D material, you wouldn't find many who can be modelled by the D&D Wizard or Magic-User.
In summation the reason why the wizard can be broad and effective, and the fighter broad and ineffective is that the wizard is a narrow concept given a broad array of tools to support that concept, while the fighter is a broad array of concepts given a small set of tools that only partially supports the vast majority of them.