Your logic is flawed, since the default (and only) campaign settings at the time was the DMs. Greyhawk didn't exist as an official setting until after the DMG in 1980-81. How could the DMG default to Greyhawk, when Greyhawk didn't yet exist? It only existed as Gygax's home campaign, just as Rob Kuntz and other TSR employees had their own as well.
No, to assume that everything had to belong into Greyhawk just because it was Gygax's home game is both nonsense and disrespectful to his campaign. This is the same mentality that lead to Greyhawk absorbing everything in 3E and the Realms absorbing everything in 5E because they were the official "default setting."
So here's the thing. If you go back and look at the OD&D and the early AD&D material, as well a Dragon Magazine, you will see the references to Greyhawk
prior to the release of the first folio. If we were to accept your logic, then anything released prior to the folio an 1980 couldn't be in Greyhawk. So ... the G Series? Not in Greyhawk. The D series? Not in Greyhawk. Acererak and the Tomb of Horrors? Not in Greyhawk. The Village of Hommlet? Not in Greyhawk. Even the Expedition to the Barrier Peaks (which is in the Barrier Peaks), and was distributed prior to the Folio, and was also a convention adventure long prior to the Folio, wouldn't be in Greyhawk.
Not to mention that there was a supplement named after the campaign setting! Yeah, it was disappointing in that it did not reveal more details, but it foretold.
Moreover, you're overlooking the peculiar circumstances that existed back then. You are also overlooking the Dragon Magazine articles that were the backbone of the hobby (especially before the post '80 explosion) that recounted Gygax's experiences in Greyhawk; I think most still remember the '77 article about Gygax and Jim Ward doing the crossover Greyhawk/Metamorphosis Alpha campaign.
So, when presented with specific names and lore in the PHB or the DMG or other books, the correct assumption was that he was using "default Greyhawk," not his home campaign, but the lightly-flavored generic Greyhawk that would later be produced in 1980 in the Folio and then perfected in the campaign setting. Again, the GH that was released was never his home campaign.
You don't have to accept that- you can pick and choose what names in the PHB and DMG aren't in Greyhawk (and other pre-1980 material), but that leads to even more bizarre outcomes.
Just curious as to when and where in the Flanaess you'd fit it in without messing up the rest of the timeline. I actually like the idea of Vecna, and I want to find a good way to insert him in.
I would recommend the views of people at Canonfire for takes that include later material- see, e.g.,-
My own GH does not incorporate any material post-'84, does not have Vecna as a God, and places his defeat (along with many of the personages that are mentioned regarding artifacts) prior to the settling of Easter Oerik, and prior to the great migration of the Suloise and Oerid people; the advantage of the original GH is that most of it is gaps to fill.