I doubt it. Why call it a "download button" if it does not download FSD Beta? If it was just an application, they could call it an "application button".
Also, this defeats the whole purpose of streamlining the process of expanding FSD Beta. I mean, the whole problem is that Tesla got flooded with applications to join FSD Beta when Elon tweeted "let us know". It does not help Tesla if the button just requests applications. The whole point is to make it easy for Tesla to expand the FSD Beta to the tens of thousands who are interested.