Noi it wasn't - it's lip synced like the rest of the promos - bit of a giveaway when they're all backing themselves in a different part of the studio with the backing vocals)
Well the Who Are You video is a bit peculiar in that it's a different set of lead vocals, lead guitar and drums (at the least). Backing vocals are the same, bass is the same (if I recall - haven't watched it in a while) but the piano might be a different take - or at least embellished a bit.
I'm speaking purely of the video that's seen in the Kids Are Alright film, though. I wouldn't have a clue if they ever synched the studio version of the song to that video, but I've never seen it done.
The Who clip was all the parts recorded live at various times during that session and then the footage was edited to make that video. Obviously that take wasn't used for the LP version. This version is much more edgey! I like the way Moon had his headphones duct taped to his head. Even the Beatles had various takes and mixes of their songs before selecting the final version for their albums.