Thing is, it kind of depends on which part of the oppression one is talking about, and which degree of transition had taken place. Obviously, societal power dynamics do not care how someone refers to themself or what someone claims with no factual backing. But the thing is, Blaire White has done quite a bit more than that. To use your example: if Blaire White had been around that time, and walked into the Bank, with her legal gender on offical documents being a woman and looking like a woman, she probably wouldn't have been given a credit line, because everyone would have perceived her to be a woman, with the legal documents backing this perception up. Now, let's say it had been Jammie Dodger ( a different Trans Youtuber, this one a Female to Male transgender , person on the left in this picture https://www.google.com/imgres?imgurl=https%3A%2F%2Fwww.thesun.co.uk%2Fwp-content%2Fuploads%2F2017%2F07%2Fjamieraines2.jpg&imgrefurl=https%3A%2F%2Fwww.thesun.co.uk%2Ftech%2F3967546%2Fhow-do-you-have-sex-trans-youtuber-jamie-raines-reveals-weirdest-things-people-ask-him-as-video-site-celebrates-pride%2F&tbnid=2d3vnKM_fv7qgM&vet=12ahUKEwjDoZvEjtvuAhUMyaQKHfiJAVIQMygaegUIARDCAQ..i&docid=xZ6qxfAIrTczZM&w=577&h=484&q=jammie%20dodger%20youtube&ved=2ahUKEwjDoZvEjtvuAhUMyaQKHfiJAVIQMygaegUIARDCAQ) who does look like a man, and who's legal documents do consider him to be a man, he probably would have been treated like a man, the fact that he was born female not withstanding.

Essentially, how society treats someone depends (primarily) neither on what this someone identifies as or what this someone is, but on what this someone is perceived to be. The exception here are issues directly related to the sexed biology, say, issues around aboprtion.