Celebrim
Legend
I'm not sure how you can square your burden of proof when it comes to Tolkien as opposed to other authors. After all, other than the regeneration, Anderson's trolls are different in some aspects than D&D trolls, yet most people would acknowledge the basis.
I think you answer your own question. If D&D's trolls have some feature that can only be found in one place, then in all likelihood the principle inspiration had to be that place. But the evidence for the Anderson troll being the basis of the D&D troll is much stronger than that even:
Here is the Anderson troll:
"He was perhaps eight feet tall, perhaps more. His forward stoop, with arms dangling past thick claw-footed legs to the ground, made it hard to tell. The hairless green skin moved upon his body. His head was a gash of a mouth, a yard-long nose, and two eyes which were black pools, without pupil or white, eyes which drank the feeble torchlight and never gave back a gleam....Like a huge green spider, the troll's severed hand ran on its fingers. Across the mounded floor, up onto a log with one taloned forefinger to hook it over the bark, down again it scrambled, until it found the cut wrist. And there it grew fast. The troll's smashed head seethed and knit together. He clambered back on his feet and grinned at them."
Here we have in one piece of text not only green skin, a very long nose, and long dangling arms - features that while telling are probably not unique to this troll - but the critical unique piece regarding the rapid regeneration of limbs. To argue against the Anderson troll you know have to show how this combination was not only not unique to Anderson, but that the authors of the D&D monster manual also had access to those other sources for a regenerating troll with a long nose, stooped posture, dangling arms, and about 8' tall that should be slain by fire. Now, I've read a lot of Swedish folk lore, and I've never encountered this combination before anywhere but in Anderson. Hence, this passage is extremely compelling evidence for the Anderson origin of the D&D troll.
On the other hand, virtually every thief of literature can move silently, hide in the dark, and climb sheer surfaces.
Yes, Jack is a computer programmer. It doesn't mean that parts aren't lifted....
No, it doesn't. But you've given zero evidence of such a lift. Find me the passage in Jack of Shadows that shares with D&D the sort of things that distinguish the D&D thief from the ordinary sort of things conjured up by the term 'thief' (prior to the introduction of the D&D thief itself).
Just like the dissimilarities in Amber doesn't mean that large swaths of the Illusionist class weren't lifted (and, for that matter, was probably why I kept playing them even though the class was terrible).
I don't necessarily disagree that the Illusionist wasn't lifted from Amber, but that's a different conversation. Finding a correspondence between specific spells in the Amber series and specific spells in D&D would for example be evidence, especially if they had similar naming conventions.