I don't think the problem is the framing of these struggles as specific or universal, it's how specific or universal these struggles actually are, and the big sliding scale there is not identity, it's class.
Both sectarians and intersectionalists privilege "identities" and the representation of those identities over the material struggles of working class people, and inevitably this comes to mean the experience of those "identities" as experienced by affluent professionals. Even when we talk about structural inequality, it always end up meaning people being passed over for promotion in an advertising firm or some other equally narrow experience, rather than anything about employment or housing or justice or education in a broad sense. Movements like Black Lives Matter have attempted to challenge that by drawing public attention back to the working class reality of most black Americans (for example), but they've only been partially successful. Liberals would still rather hear about how the Grammies are racist because they only gave Beyoncé a seven golden scepters this year than hear about the water crisis in Flint or about violence against immigrants.