It seems to me that a game design that cannot accomodate situations that will come up frequently is insufficiently robust.
Who defines "frequently," the WotC extensive pre-3e market research? Or you?
If you want a game with the power to handle every likely situation that could crop up and do it in a universally fair and equitable manner to all PC's and NPC's, you want something unrealistic. A simple *game* cannot accomodate every frequent situation. Heck, MONOPOLY gets hazy when you start throwing the piles and piles of house rules into it, and Chutes and Ladders doesn't tell you what happens when you want to go off the board, and Poker can be played a million different ways that are all unfair to someone. These don't tolerate every frequent occasion. You shouldn't be able to play Go Fish when you're playing Poker.
Having a norm (that is sufficiently common) allows a game system to deviate from it and be accountable for why things are happening the way they are. And since every DM is a bit of a game designer, a DM can look at a particular encounter and ask "why didn't this go well?" and come at an answer derived from what his party is facing vs. the assumed norm. That is a baseline, and a baseline that it is assumed you will deviate from and toy with yourself, but a method of measurement.
D&D does not need to be designed for "PvP" play to be a well-designed system. Quite the contrary, if it *was* designed for that style of play, it wouldn't be a game about a team of adventurers going into dungeons to beat up monsters and take their stuff -- it would be a game about fighting each other. There's no reason D&D has to be balanced like that, no matter how common the occurance is. No matter how many times people go off the board in Chutes and Ladders, you do not need to design Chutes and Ladders to accomodate going off the board.
Rather than change D&D, you're better off finding a new game tha plays the way you want, like the D&D minis game, or a CCG....these facilitate head-to-head play. I can't think of any d20 system game that does do that, either, but maybe that's because most people who play aren't really interested in that style, or that the other games like to follow too closely to the D&D template....anyway, D&D really doesn't enable you to do that. And needing magic, needing a cleric, needing a fighter, needing a rogue...these are not flaws, they are features that enable a group of people to work together against an antagonist, each adding their own particular dimension to the team.
That's really poor game design if you have that many restrictions on what you can play and still consider it balanced.
You seem to misplace the goals of the game design, I think. The classes and PC abilities were designed so that every person who was a player could contribute against a common enemy in a meaningful fashion. They aren't made to be self-sufficient, and they offer no claims to it. If you don't have someone who can heal (which other classes can do, in different ways), you're lacking part of the team, just as if you don't have a front-line fighter, or a skillmonkey, or a blaster. That's actually the *opposite* of bad game design -- the designers wanted people to play a diverse set of classes, so they made it an essential component to the game's assumed level. This way, not everyone is playing a fighter and feeling homogenous and mundane, but every time someone plays a fighter, they can play a different kind of fighter -- you could play a Fighter, or a Barbarian, or a Str-based Monk, or a Cleric of War, or an aristocratic Swashbuckler, or an intimidating Samurai or a wildernes-loving Ranger or an evil-smiting Paladin or an accursed Hexblade...these can all hold up the proper role in the party.
But if the party is made up entirely of those types of characters, it will be unbalanced. And that's not a flaw...that's the game saying "putting all your eggs in one basket is a bad idea" or "diversity is essential for success" or "just like in real life, no one can do everything by themselves -- you need friends and allies." This encourages team play, which is the express purpose of the game.
A well-designed game is a game in which the experience of playing the game by the rules facilitates fun in the idiom of the game's express purpose. Monopoly is a well-designed game because playing Monopoly by the rules is a fun expression of buying and selling your friends. Poker is a well-designed game because playing Poker by the rules is a fun expression of the power of chance and the ability to discern the motives of others. D&D is a well-designed game because playing D&D by the rules is a fun expression of getting together with some friends to beat up bad guys, save the world, and change over the course of many playings. It doesn't *need* to help you play an all-Fighter-and-Rogue party any more than Monopoly has to help you play without money. They both can and are played these ways, but it's a different sort of experience than the designers intended, and it's not poor design just because they don't meet your every demand from how to play them.