We should definitely not put a Shaivism tag on a question just because it's about Shiva. Questions about Shiva, questions about Shaivism, and questions about the Shiva Purana are three conceptually distinct categories.
Vaishnavism and Shaivism are philsophies that, as you said, respectively posit that Vishnu and Shiva are supreme. Now it's true Vishnu Purana focuses on the greatness of Vishnu and the Shiva Purana focuses on the greatness of Shiva, and but that's not all these Puranas are; they are illustrious accounts that cover all sorts of subjects. And by the same token, while Vaishnavites might put more stock into the Vishnu and Shaivites might believe the words of the Shiva Purana more, the differences between Vaishnavites and Shavites doesn't boil down to the differences between the Vishnu Purana and the Shiva Purana. Some disputes that Vaishnavites and Shaivites have may be about things that have nothing to do with the topics discussed in those books, and other topics may be addressed in those books but the determinants of the beliefs concerning such subjects may lie elsewhere, like the Pancharatra Agamas.
So to sum up, we shouldn't conflate Vaishnavism/Shaivism with Vishnu Purana/Shiva Purana. The first two tags should be used for questions about the philosophies, and the second two tags should be used for questions about those particular books. If you happen to want to know what the story of Kala Bhairava is in the Shiva Purana, that doesn't mean you want to find the "Shaivite" view of Kalabhairava. And if you want to know why Gaudiya Vaishnavas place more emphasis on Krishna than on other purna avataras, the answer is not going to be found in the Vishnu Purana's description of Krishna.
