I don't think a fighter should be anything. The existence of the class seems of questionable value to me, because while fighters need abilities to be useful, balanced and interesting, whatever abilities you give them will always cause them to break from somebody's conception of "fighter." The issue is, in essence, that fighter is too broad of a concept. Better to have a variety of class options, each with a clear set of mechanical concepts in play, and avoid calling any of them "fighter."
Now, what should a wizard be? That seems like a contingent question to me. My first instinct is that the answer is pretty much the same — the reason wizards have to be too powerful is the same as the reason why fighters have to suck. Both concepts cover too much territory, save that one covers it by exclusion and the other by inclusion. Both need to be broken up into smaller, better-contained units with a clearer scope.
QFT. If there is a fighter class, then it should be broken up into multiple styles and archetypes, and there shouldn't be a "basic fighter". There is the zweihander duelist, the two-handed axe/hammer brute, the polearm master, the sword-and-shield defender, and the axe-and-shield vanguard, and they all share the same chassis (BAB, saving throws, hit dice), but there is not a "basic fighter". Those archetypes I listed can come from the current fighter with a specific selection of feats, but that shouldn't be the case. You could just as easily take Power Attack, Weapon Focus: Rapier, and Improved Unarmed Strike, but that shouldn't be from the class. Sure, with the new fighter, you can take the Vanguard archetype and take Weapon Focus: Rapier for style points, but you won't be as effective as if you used an axe, and most importantly, it tells you so. You can be half-decent, you'll still get the Vanguard's speed boost and charging features, but you won't get the +attack bonus you would've if you'd grabbed an axe.
Ultimately, though, I dislike class systems for anything other than quick character creation for a casual game.