But this is presented as a "story", an internet back-and-forth. The girl up top started off by accusation that it was a "privilege", and she was wrong, but she used the term bc she thought she was right.

From her childish logical POV, men had that "privilege" above "normal" - her normal. Men had that "special right", that "advantage", that "immunity", granted only to the particular group of humans in her world who are men yet denied to women. So it wasn't her word choice that was wrong - the word accurately described her childish way of looking at the world.

She was, however, wrong. Sort of. Mostly. Bc while men grant that non-special right to everyone regardless of gender, women only grant it specially to men (and not even all women do, it's a special kind of outdated Victorian cultural attitude that does so).

So what I am saying is that the word "privilege" was correct here... not in spite of but because it is wrong -> it is used to show how wrong the underlying concept is.