When the DMG talks of game balance, it doesn't mean anything at all like the term 'game balance' is used today. The idea of parity between the classes or even characters wasn't even a consideration.
When 1st edition AD&D talked about balance, what it had in mind was that the game attained an appropriate degree of challenge and reward so that it was always still a struggle to attain further success but never so much of a struggle that success felt arbitrary or forever out of reach. When 1st edition AD&D talks about balance, it has in mind the idea of 'fairness', but it doesn't remotely care about 'equality of results'. It only means that each player has equal oppurtunity to succeed. That his character may be inherently weaker than another player's character at the table is not really a consideration. Over time it is assumed he will have a chance to play many characters, some of which will obtain great power and some of which may become favored and successful despite the early odds stacked against them.
Balance as we would use the term today is barely considered.
This manifest in everything in the game.
Players rolled for their stats randomly. This was intended to be fair. It was never intended to be balanced.
You might get better than a 16 in your prime requisite, and so advance in levels faster. This was intended to be fair in as much as the player was playing a character with natural advantages and so should do so. It was never intended to be balanced.
You might have such bad stats that you could only qualify to be a rogue or perhaps a M-U that lacked the potential for much advancement, or you might have such good stats that you could play a Ranger or a Paladin. This was considered to be fair, in as much as any one might someday get a character of this sort and would then qualify for the benefits, but it was never considered to be balanced as we would use the term.
Someone criticized this philosophy as an attempt to balance things as a result of 'rarity' and mocked the effort as stupid. And it would be, if to Gygax balance meant what the mocker means. But it clearly doesn't. Gygax believed the game balanced if superior advantages were rare because he wasn't meaning a balance between players as if the central idea of the game was a spirit of envy and competition between the players. This was not even really considered and would only have been considered if the focus of the game was combat between the players. Had this been the focus of the game, Gygax, with his war gaming background could certainly have produced rules that granted an equal likelihood of victory to both sides, but it was never the point of the game. Rarity balanced the game precisely because the presence of one more powerful player did not itself it the balance between challenge and reward such that the campaign became imbalanced. And, if it happened that the players managed to all acquire rare and powerful abilities through fair play, then that was their just reward for long and diligent play.
How can you complain that the system was broken because people circumvented the system (or to be more clear, they cheated) even when they had been expressedly adviced not to do so.
So to answer the question, "Yes, the system was designed to be balanced.", but for the most part, that didn't mean then what it meant now.
Actually, Gygax probably succeeded at his goal of 'balance' far better than any modern designer has succeeded in theirs.
