Is armageddon really fair for white?

Especially at the higher levels, GMs playing black can just play defense and not have to look for a winning position.

What I propose is a baseball extra innings approach. Both sides have to play white. So, you would have two armageddon games per extra inning. If white wins the first, then the other player has to win like the rules are now. However, if they draw then they have to do 2 more. 

Black could also try play for a win too, making their time as white easier. If they draw or win when they are white then they win the match.

I've always been partial to the idea of bidding to play black. For example:

Player A: I'll take Black and 4 minutes

Player B: I'll take Black and 3.5 minutes

Player A: I'll take Black and 3 minutes

Player B: I'll take Black and 2.5 minutes

Player A: Okay, you play Black.

But what's the attraction for only one game? We don't do that with classical. We don't do that with rapid. We don't do that with blitz. Why only one game with armageddon?