You are forgetting the section on page 7 of the PH, or is it something that you call "half-baked" ?
As a role player, you become Falstaff the fighter. You know how strong, intelligent, wise, healthy, dexterous and, relatively speaking, how commanding a personality you have. Details as to your appearance, your body proportions, and your history can be produced by you or the Dungeon Master. You act out the game as this character, staying within your “god-given abilities”, and as molded by your philosophical and moral ethics (called alignment). You interact with your fellow role players, not as Jim and Bob and Mary who work at the office together, but as Falstaff the fighter, Angore the cleric, and Filmar, the mistress of magic! The Dungeon Master will act the parts of “everyone else”, and will present to you a variety of new characters to talk with, drink with, gamble with, adventure with, and often fight with! Each of you will become an artful thespian as time goes by — and you will acquire gold, magic items, and great renown as you become Falstaff the Invincible!
So why I agree that there is that aspect of "playing a role in the party" according to class, there is also the part about acting in character, and this is reinforced by the descriptions from the modules that I have also provided.
The notion of party role was heavily reinforced in 4e, though, so it's not a general trend, more like specificities of editions, the way I see it.