That's because it's almost certainly true! it's why Indians speak Indo-European languages, and why the same Y-chromosome haplogroup (R1a) is common among (for example) Indian Brahmins and Norwegians.
> This theory is often connected with a view that fairer-skinned people represent "true" Indian culture
The Hindu religion in its most ancient form (see e.g. the Rig Veda) derives from the Proto-Indo-European religion, parts of which have been reconstructed: https://en.wikipedia.org/wiki/Proto-Indo-European_mythology
(Obviously modern India culture comes from a lot of other sources too!)
> including the Aryans who are said to have invaded India.
You phrase it that was as if to imply that it didn't happen. But I think the evidence that it did: linguistic, genetic, archeological, is very strong.
Actual Indo-Europeanists today eschew the label "Aryan invasion theory" in favour of e.g. “Aryan migration theory”. While genetic evidence makes a (limited) case for presence of force and population movement in the spread of the Indo-European languages to India, the bulk of the spread was through language shift whereby the indigenous inhabitants gradually adopted Indo-European languages as higher prestige.
1 - https://en.wikipedia.org/wiki/Bactria%E2%80%93Margiana_Archa...
Indian gods and historical figures are usually depicted as fair, in their own, contemporary art.