It's important to acknowledge one's advantages and disadvantages in life and be aware of unconscious biases and systemic issues. I think the term "white fragility" is going to piss people off just by the way it sounds, though. There must be a better term.
I agree that "white fragility" is not a great phrase, and I think it only really works with people who are already "on board" in some fashion (primarily LGBTQ+ white people, or just white people who already consider themself "woke").
I suspect that white people like me could "grease" the situation a bit by doing more to even the load. By that I mean to use our white privilege proactively. A racist white person is probably more likely to take the words of another white person more seriously, even when the phrasing is more eloquent, interesting and nuanced coming from a non-white person.
As I (clumsily) expressed in another thread about racism, white people are a bit unique in the way we can "peace out" if race-discussions ever turn too heated, or too pointed/confrontational.
If I wasn't white, I think I would probably be a little wary of white allies, simply because there is always that "emergency-exit" available to us if things really get awful. Dealing with race-issues is an active, deliberate choice for white people, and opting in - as opposed to just being forced to deal, like every non-white person has to - is going to impact how seriously each person interacts with an issue.
I think - for similiar reasons - that LGBTQ+ people (like me) have a less robust "trust in another stranger like you" because a lot of LGBTQ+ people has the option to "peace out" by conforming to society's norms ("acting straight"), and become invisible in a way that isn't possible for people who don't pass as non-white, in terms of being identified as non-white.
I think I fucked up in that other thread by not properly explaining where I was coming from, so I'm a little "on-guard" when writing this. But, still, I do feel that there is some merit to what I'm saying.