Hinduism should in fact simply be generalized as "Dharmic Traditions" instead. I think that would also reinvigorate the philosophical traditions, and ideas that

Hinduism should in fact simply be generalized as "Dharmic Traditions" instead. I think that would also reinvigorate the philosophical traditions, and ideas that each region of India has. However with that said you're also ignoring the very real fact that a "Hindu" of Kerala, Adi Shankracayra, is in fact a pan-indian philosopher worshipped from North to South, and from East to west. He in fact established pan-Indian/pan-dharmic institutions all over India from the south up-to Kashmir, to both the western and eastern parts of "India". This pan "India" narrative also plays a huge role in Indian mythology where mythological characters like Augustya rishi are worshipped, or at the very least understood as great human beings, all over India among the "Hindus". Not to mention the monuments, places of worship, and religious features that Hindus believe, similarly, have a pan-"Indian" presence. The problem is that Indian politics is fundamentally based in the idea of Kautilya's Mandala politics, and saptang rules which take into account variable relationships, and is rather less reductive/zero sum in comparison to Machiavellian strategies. This political strategy invariably also plays a part in the spread of various Dharmic ideas, and institutions.