I'd certainly consider the Player's Handbook a core product. And as you can see from my post, it clearly states that role playing is a thing.
the 1e PHB the importance of role playing and imagination
<snip>
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 "godgiven
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 Folstaff 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 ortful thespian as
time goes by - and you will acquire gold, magic items, and great renown
as you become Falstaff the Invincible!
The proper interpretation of that passage has been discussed at some length in
this recent thread.
Nowhere, anywhere else, does Gygax's PHB talk about playing a PC, in the sense of developing and expressing a distinct personality or characterisation. The advice on pp 107 and 109 of the PHB about how to prepare for a session assume that the only considerations are efficiency and effectiveness across the party (there is not the least hint there that, for instance, the Mistress of Magic might refuse to memorise fire spells because "that's not what my character would do!").
And when, in his DMG, Gygax talks about good roleplaying (in the context of experience points and level gain), it is all by reference to class functions, not to character personality. (The nearest thing to an exception is the reference to alignment.)
At the time the 1st ed AD&D rules were written, I don't think the books were talking about roleplaying in the sense of creating a character's personality. Whereas the 2nd ed AD&D PHB puts this front and centre. The contrast between the way the two sets of books present the point of the game, and what it is to play a PC, is very marked.
That's not to say that people weren't doing it the "2nd ed" way before 2nd ed was published. The 2nd ed rulebooks were following, not leading. But I don't think the 1st ed PHB addressed this issue of PC personality in any serious fashion.
Similar passages exist in both Moldvay's and Mentzer's basic books as well. Mentzer is way obvious and they have a whole page devoted to it on page 2. Moldvay says a couple times that you take the role of your PC to create the story, and in the example D&D session, it clearly has the role-playing aspect as important.
I don't agree with your reading of Moldvay. (I've never read Mentzer.)
For instance, when Black Dougal dies, all that happens is that one of the other PCs takes his pack and the cleric player says "I give him the last rites of my church." There is no genuine sense of PC personality. (Again, with alignment as an exception in the context of Morgan Ironwolf being healed by the cleric.)
Contrast the 2nd ed PHB, which talks about creating a unique and memorable personalilty for one's PC. Nothing in Moldvay says anything like this; in this respect, Moldvay resembles Gygax's PHB.
EDIT: The importance of
imagination in RPGing is independent of the idea of roleplaying in the sense of character personality.
Because what distinguishes a RPG from a boardgame is that the fiction matters to action resolution, imagination is absolutely crucial. Eg in the example of play in Gygax's DMG, the players come up with ideas like using a pole to poke in the stream, or forming a human pyramid to investigate the holes in wall that lead up to the secret door with the ghouls behind it.
But - as the Gygax exmaple itself shows, with character absolutely devoid of roleplaying in the characterisationse sense - you can exercise this sort of imagination without developing a distinct or unique personality for your PC.