It isn't that people don't care. It's that public sentiments about 'beauty privilege' can't be used to mobilize social or political movements. White privilege, for example, is used by the left to define their ideological position and make specific policy recommendations. By contrast, what political project would ever take up 'beauty privilege' as part of its platform?

Out of a nearly infinite number of real phenomena, the social consciousness picks only those that suit particular purposes. 'Beauty privilege' is a brute fact of life, as you describe so well in this article, but the reason why society doesn't focus on it is simple: doing so would serve no social or political purpose.