They take it as a dismissal of their ideas, not necessarily of their identity.

"No matter how thoughtful your opinion that differs from mine, I'm going to regard it as invalid because you're white" is how it comes down to many people. Which side is the one with the fragile ego there? The one that takes offense at having their ideas dismissed on such a basis, or the side that has to use such an excuse to not listen?

The point of "because you're white" is, you have no personal experience in that matter. Despite an obvious lack of expertise, privileged people (and I consider me part of that group) feel that their opinion is of at least equal value, and that it should be listened to.

Personally, I quite easily give my opinion unasked (like here), and do feel rejected if it isn't listen too. Likely, because I grew up the way, that my opinion matters.

I would imagine, the experience would be quite a different one, if your thoughtful opinion would be voiced after being asked for it.

Those are fair points.