But that's the problem, isn't it? Humans mostly aren't up for balancing things so delicately. It's possible for us to be delicate, but it's not robust and reliable.
The intuition of the liberal enlightenment thinkers was that we needed to stress our commonalities as humans and giving each other space to be our unique selves, because the errors of excess were mostly in the direction of intra-tribal conformity and inter-tribal strife. These intuitions seem especially apt given our modern understanding of human psychology.
Christians are supposed to love their neighbors as themselves. They also have a disturbing tendency to support abusive behavior to, for instance, prevent unauthorized immigration, especially if the people trying to immigrate don't fall into the same identity groups.
As if this didn't make it hard enough, intersectionality has mostly been introduced through a critical theory lens that provides numerous ways to avoid evaluation of whether ideas are true and courses of action are useful. It sets up an unassailable platform from which to critique the ills and injustices of society, but refuses to check if the platform itself is rotten or badly situated. This makes it especially hard to winnow out good ideas from pernicious ones.
And if you stir this into the pot of social media, where engagement bests delicacy and balance is viewed as tepid and inauthentic best, one wonders how this could possibly work.
It's all well and good to use the idea of intersectionality (or the older, more general, and more powerful idea of statistical covariates--but try explaining covariates to your average judge and you're not likely to get very far) to help find patterns of harm or mistreatment that we might otherwise overlook. These harms were substantial and some are ongoing, and it is important to find them and address them. But, given human nature, it seems that we should treat the intersectional lens almost as if it were the One Ring: its tendency is to corrupt and cause harm. Bring it out in the courtroom in key lawsuits, then take it off your finger and leave it on its chain!
Indeed, some of the postmodern thinkers, such as Marcuse, who have gained favor among those who lean towards an intersectional perspective were open about wishing that people would find society (meaning each other!) intolerable and imagined that out of the ashes would, somehow, come a glorious, just, equal society.
But if you look across history, out of the ashes of a society that has turned on itself come people who have been badly, badly burned.