You are both doing the same exact thing at the same time, there is no strategy other than 'don't forget up'. It can't even be judged like traditional dance or music contests, as those rely on contestants playing a wide range of different music or different routines.
That's the same logic as saying there's no competition in LoL because the objective for both teams is to kill the other team. Are both competitors supposed to do something different? That wouldn't be very fair, would it?
And yes, the fact that many tournaments revolving around the game today exist
should tell you that it can be judged, regardless of whether or not it's "traditional". If you're not going purely off of the highest score, you could perhaps have a "routine" competition; who can make the best "actual dancing" routine to whatever song they please. This kind of competition is common in Pump it Up.
Even then, there are plenty of gamemodes, too. Cooperative gamemodes, in fact, and as I linked in the video, game mechanics that can make any competition interesting and dynamic.
Also, why would a player need to live here? It's not a team game.
There's a team-based gamemode, and a cooperative gamemode.
And if there were to be more interest in the game, the same as the amount of interest that LoL seems to have built up, giving the same treatment to osu! as was given to LoL would make sense.
but no one plays it competitively.
I honestly hope you're not being serious here.
It seems like you pretty much don't know what the hell you're talking about when it comes to osu!, or rhythm games in general.