Gygax's AD&D is a pretty solid game.
The books sure were. My DMG is battered but still in one piece, I can't say the same thing about all my later-era D&D books.
Yet if everyone is equally powered in dimensions X Y and Z what do they need the rest of the party for?
Neither over- nor under-powered does not mean exactly equal nor identical. A sufficiently overpowered (particularly if also overflexible) option renders others non-viable 'traps.' So if you have many balanced choices along each dimension or within each pillar (or pick your terminology/metaphor), you can have many unique characters who are balanced with eachother in each of those areas, and the GM can emphasize different aspects of the game without worrying about sidelining anyone unfairly.
The whole point of the party concept...
I'm not so sure there /is/ a "whole point" to it, I think it just sorta happened. If we go all the way back to what's public knowledge (or rumor) about the earliest D&D games, there were a lot of wizards being played, and someone wanted to play a vampire - and someone else a Van Helsing type (and got the Cleric).
I suspect the main point of the 'party' is to accommodate multiple players. It's also the main point of balance in that context. (Though, in a single-player game, there's still a point to balance.)
Early D&D was imbalanced in some ways, sure, but in others it had balances the later editions could take a lesson from.
Mostly in the sense of learning from the mistakes of history, sure. ;P
But,
Casters of all kinds, for example, were greatly kept in check by their being so easy to interrupt. No such thing as combat casting, and spells took time during which other people could mess you up. Later editions (most notable 3e) took away those headaches and had/have no end of issues with overpowered casters.
There has been a steady trend away from restrictions on casting, and it seems like the impetus wasn't to make casters more or less powerful, though it does the former quite noticeably. 3e loosened up restrictions on casters substantially but barely reigned in the power of spells, and full casters ruled Optimization Tiers 1 & 2, possibly overpowering and overshadowing everything else. 4e /further/ loosened up most restrictions on casters, but more or less balanced them by reducing the power/versatility of spells, and greatly increasing the peak power/flexibility of non-casters. 5e further loosened up restrictions on casters (not that there was a lot left to loosen) and partially restored the high-power/flexibility of casting.
So even though spellcasting restrictions have been on a straight-line road to oblivion, balance has not tracked them.
The staggered level-advancement tables, where Thieves bumped almost twice as fast as MU's, were also a balancing influence.
....
That's a bit of a myth. When Thieves hit 11th level, M-U's hit 10th. In typical games, you never had a separation of more than 1 HD with M-U's, and no more than about 2HD with Paladins and Barbarians. It gets slightly better after that, and you might get separation of up to 3HD later on, but in 1e after name level the benefits of increasing level were very small for non-casters (and thieves especially).
Nod. Double the experience meant about a level behind until name level, where it prettymuch stopped mattering for non-casters. It was a surprisingly neat system: it helped minimize the consequences of PC death (your character died at 5th, your new 1st-level character would be 6th by the time the highest level PC was 7th or 8th); kept the MCing system basically functional (in terms of hps & saving throws, for instance); made dual-classing remotely practicable; but it didn't much 'balance' classes with different rates of advancement.