Surely Bladesinger's should get the Acrobatics skill rather than Performance.

Not really. The idea of bladesinging is to use magic and dance to your combat. You move and attack with a blade as if dancing to a song.

Performance is the dance skill. That's why you get it.

100% that.

Yeah fair enough.  It does say "dancing to a song".  I just don't envisage the Bladesinger like that, I see them, yes you could call it dancing around, but I see them as moving about in a speedy athletic way rather than to music.

I mean, I get that. I had a bladesinger that was focused as a duelling champion. But instead of bladesong being "dancing", it was more like a "battle mode", with glyphs appearing on him and granting the benefits that way. He did not learn it as a tradition, it was a result of experimental magic he did to himself, in effort to make him a better duellist. I justified the performance skill as "showmanship" like playing up a crowd, impressing onlookers with his prowess of sword and magic. 

👍👍

I flavored a Bladesinger once as lightning themed, and his Bladesong was bolts of electricity sparking off of him and grounding into the earth around him, creating notes, which he would then make music like this.

And rolled performance to see how good it sounded.

Pure fluff, of course, but was fun.