It is well known that a lot of games in the top level round-robin tournaments end in draws, often not very exciting ones. Many consider that detrimental, as it leads to spectators and sponsors losing their interest in the game. The anti-draw measures that have been tried are Sofia rules (ban on draw offers) and/or Bilbao rules (3 points for a win, 1 point for a draw).
There also has been a proposal to introduce the following rule: if a game ends in a draw, the players switch colors and play another game with faster time control. If that also ends in a draw, they switch colors again and play one more game with even faster time control. And so on until the first game won by one of the sides, who then gets the full point. The tie-breaks may be played after the round (which shouldn't be too bad with modern time controls), or on separate days like the adjourned games of old times.
This leads to every pair in each round having a winner, and, at its face, this has several immediate advantages. The pre-agreed draws, or just quick draws between players who are not in a mood to fight, would not make sense anymore. A player with white pieces gets an incentive to try and get as much winning chances as possible, or face a rematch with black; playing solidly for a draw with white against a stronger opponent would not make sense.
However, despite all these advantages, and despite the fact that there are many rapid/blitz tournaments, I have never seen this scheme implemented. Why is that? What are the downsides of this rule? Have any top players ever commented on that?