I think it is relevant. It is my understanding that a privilege is something that is unearned. Someone is not being an inconsiderate asshole just because they've watched their metabolism, & as a result, they can fit in their fucking bathtubs.
I'm not saying that just anyone can lose weight like nothing, or that fat discrimination does not exist, but I still don't believe there's any such thing as "thin privilege," & if there is, it's definitely not any of this "I have to pay extra/can't fit stuff." This always seems to come down to logistical reasoning, like more difficult things cost more to do, & that at certain points, we can't feasibly design things that accommodate people of a certain size.
It's also significantly more complicated than "fat people are oppressed," because if you say that weight should never be a concern, that's basically an outright lie. You have to balance the idea of body acceptance with the idea of proper weight management, & that's not easy to do. Also, unlike white privilege, or Christian privilege, or male privilege, you can be fucked sideways by being thin, too.