Lyxen
Great Old One
Are you even familiar with the word "premise"?
Merriam-Webster: "a statement or idea that is accepted as being true and that is used as the basis of an argument", for example "There are in general gods and goddesses in D&D editions, settings and modules", which is absolutely proven despite you finding a few sentences out of the hundreds of thousand in D&D publications that show that yes, there are minor exceptions.
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Nary an Immortal in sight.
Well, maybe you should read more of Basic/BECMI than just a single page (which again proves that what you think are EXCEPTIONS), because the rest of BECMI clearly explains that the faith is placed in gods/goddesses and that there are churches with their hierarchy.
Molday Basic:
Expert Set:
Companion Set:
And there are so many mentions of gods in modules B/X that I won't bore you with them, but for example:
1978 - B1 (In Search of the Unknown) - The dungeons has a worship area with a horned, evil-looking, idol (surrounded by runes and glyphs and with a sacrificial pit in front of it) as a "token gesture to the gods".
1979 - B2 (Keep on the Borderlands) - The keep has a chapel, although no details of the worship there are given; and the Caves of Chaos contain an evil temple that appears to be run by demon worshippers.