This is an excellent, thought-provoking article! I think you're exactly right about the fundamental contradiction within multiculturalism, which yields incoherent outcomes like simultaneously decrying cultural imperialism while insisting on using "Latinx" to describe people of a culture whose language is butchered by trying to use such a term; or screaming loudly about how the white patriarchy is oppressing women in the U.S. while remaining curiously mute about how the Islamic patriarchy is oppressing women far, far more in many Islamic areas (including sometimes in the U.S.!). Because of the internal contradictions, you can't guess how these things will turn out: you just have to watch how the wind is blowing and follow the crowd.

And I also think you're on-target that thinking about obligations / responsibilties / duties is a way towards a resolution. Indeed, I think we always need to consider such things when thinking about so-called rights, because a right is only a right insofar as we are collectively willing to accept the necessary obligations to uphold it. I see no reason why this should be any less true of cultures than of individuals.

A good discussion about the interplay between rights and duties can be found here: https://bostonreview.net/articles/samuel-moyn-rights-duties/