I don't even like the term "white fragility", because them being shocked/saddened/appaled by the discussion of the topic of racism (when they're not being consciously blind to it) itself comes from a position of power. They can afford not to care about racism, because they aren't affected by it. This goes also for most "progressives" and whatnot, who just by their bieng white can read/inform themselves all they want about racism, but whose structural power shields them from it. Hell that's even the case for some advantaged Black people who may not as exposed to racism than less priviledged...until they get their wake-up call.
So the term itself sounds problematic to me. I know it has a sense of "mocking" such "fragility" but in some twisted it compounds the idea of the "innocent white" who "just never thought about it". Well, if you never thought about it, that's white privilege, which comes from white supremacy, which is anything but "fragile".
Also, it must be known that tears and emotions can also be political. Showing tears and whatnot will almost automatically bring empathy and sympathy, and now all of a sudden whoever is talking about racism might be the bad guy who'se making a "fragile" white person cry or feel "uncomfortable".