He did good work nobody is disputing that. If his work sucked it wouldn't be published in the Annals. Again, there are only two possibilities:
1) Either even top people in algebraic geometry (e.g Scholze) don't care enough about birational geometry to know who the supposedly "leading" people are, but in this case there should have been no medal in "birational geometry"
OR
2) They care about birational geometry (I think they really do) but Birkar was simply not the leader you are claiming he was. (On the other hand I am sure Scholze heard of Hacon).
There were many other candidates who were at least as much deserving as Birkar who didn't get it, e.g Williamson or Wei Zhang or Brendle. But they went for Birkar and it's hard not to think that it was not political. There was Mori of course but the refugee angle is also hard to overlook since it was emphasized. Had they given it to him without rubbing in everybody's face that he's "one of those refugees" we wouldn't be thinking this.
Did PS literally say he hadn't heard of Birkar? Or did he say something along the lines of not being very familiar with his work?
If someone cares about birational geometry and cares about Hacon's work, they would know who Birkar is, since Birkar was a coauthor on Hacon's biggest paper. You have yet to address this. You regard Hacon as a leader in the field. Well, Birkar was a coauthor on Hacon's biggest, most impactful paper and additionally did very important work without any coauthors. What does that make Birkar?
Who rubbed in the "refugee angle" in people's faces? It wasn't the mathematicians who awarded the medals, but the media. The media has to find something nontechnical to write about each of the medalists. I saw a video that focused on Figalli's long distance relationship, for example. Who cares.