Firstly, the Pope relocating to a different holding wouldn't change the primary culture of the Papal States, but the event does. That is why I always thought of it as another theocracy taking its place instead of relocation. And the Donation of Pepin is just a document. Like the treaty between Rev. France and Austria that gave Tyrol to Bavaria was just a document. As a grand historian, you will know how happy the population was to think of themselves as Bavarians there. I don't want to say the Donation wouldn't legitimise the Pope as the ruler of these lands, after all, its whole purpose is to confirm just that.
I'm just torn when it comes to the game's interpretation of primary tag:
On the one hand, I can see that losing Rome and surviving in Romagna or Avignon and then getting back their former land via defection would make sense for the Papal States, more so than the provinces going to Urbino.
On the other hand, let's think of a more bizarre scenario:
A primary tag would only have any meaning if anyone held Umbrian land without being of Umbrian primary culture, so let's imagine someone who didn't give two shits about the Pope and his right to Rome would annex him. Someone like Tunis. The Papal States then reemerge in Salzburg (or any other Germanic HRE landlocked province, which is by far the most likely outcome). Tunis gets wrecked by rebels and Umbrian separatists defect to the Papal States. Now an Austrian (any other Germanic) cultured, probably landlocked, probably HRE member OPM would get Umbrian land, and unless the Pope gets a specific event to relocate its capital to Rome, it cannot relocate its capital from inside the HRE out of it. And I don't think the AI is smart enough to leave the Empire in such a case.
I think that was the main reason why the Pope is not primary tag for its culture: The Papal States have some exceptions and special rules that are no problem for other countries, but since they are special, difficult situations could occur. So to sidestep something that could screw up their system a little more, Paradox just went for the other option of using another tag.