Unearthed Arcana is mostly a collection of articles (some of which EGG wrote, to be fair).
I believe he wrote all of what actually got published in Unearthed Arcana. I could be wrong on that, but there were a lot of stuff from Dragon by other authors that was more widely used in the games we played and more often revisited in the magazine, but did not make the official rules. This includes off the top of my head better Monk, better Bard, different Ranger updates, different Paladin updates and better unarmed combat.
I'll agree that what we'd now call the core 3 are very clearly Gary's writing (so much so that they are notable in their use of 'Gygaxian prose'). Honestly moreso than oD&D, which feels more like a (n incomplete) summation of playtest findings collected by Gygax.
I do think there's more nuance in the statement that we shouldn't treat the person and the edition as the same, though. Gygax contained multitudes, and only himself as a point in time truly aligned with 1e AD&D -- and particularly the AD&D that existed flowing from his pen (/typewriter) as he was writing these books. Even during the 1e era he vacillated wildly in thoughts on the importance of DM impartiality and other important game qualities.
I agree with all this, but the core rulebooks of 1E AD&D are Gygax's and in that respect the edition is as well as those are the defining manuals. The other pieces of it you describe are minor
Ben Franklin changed his opinion and writing wildly over the course of his life on a variety of topics related to liberty, freedom and religion. Yet all his writings are still his writings.
Where I shift back to OP's point, is the use of Gary in support (or just framing) of anything else (usually an argument in favor of/against something, or in how something ought to be).
I agree with the OP on this, however referring to an edition is a pretty broad stroke, much more so than the much narrower definition you are talking about here.
Also as I stated he is IMO essentially the core of 1E AD&D, not part of it, not the main contributor, the literal author. This is not true of later versions or oD&D or other earlier or contemporary versions outside of 1E AD&D specifically.
I don't see any utility in using Gygax name to support or frame a position. In 1E AD&D sure, you can say "this is the rules" because he wrote those rules, but outside of that it is pretty flat as when you move outside what he actually said in those books you run into the wide variations you talk about.
The most obvious (in my mind) case would be arguing for something to be the way to play D&D 'as Gary intended it.' This ought immediately call up the two responses of 1) 'how do you figure?,' and more importantly 2) 'so what/yes, and?'
Any argument about 1E using Gygax to frame a historical point of reference is fine, but 1E had a ton of shortfalls and downright contradictions written by his own pen in the core rulebooks.
Whether you like Gary or not, his version of AD&D was heavily flawed. Any play of 1E AD&D necessarily requires houseruling or deviating from what is written because what is written (presumably what he intended at least when he wrote it) is either unplayable or in conflict with other things he wrote in the core rules.