"White privilege" is one of my most hated phrases, not because of the phrase itself, because of how it's typically used. Are white people generally more privileged than minorities? Sure. Can you just assume someone is privileged because they are white? No. That's where racial insensitivity comes in. If you know someone who is white, know their background and they were privileged, then ok. But to just sling that phrase around to people that you have no knowledge of their background, upbringing, etc is just rude, obnoxious and flat out nasty.
I think it's just generally used as a term to throw at people to belittle them for not agreeing with your political and social views. "Well, you're white and male, so you can't really comment on the matter" essentially. It's just a cheap and lazy way of avoiding debate and propping one's self on a pedestal and trying to make someone feel guilty for the way they were born. I guess responding in a manner like "I'm sorry, respect your opinion and all, but I don't think you understand the plight of trans people and how they feel they are unfairly targeted by cops" is too civil these days.
In this instance, I think it was poorly used. Not sure I'd call that woman a racist or a sexist, but it at the very least was insensitive and rude of her to say that to this guy for merely asking a question. Not sure it was suspension worthy on either side. A handshake and a couple I'm sorry's probably would have sufficed.