What are the downsides of fast tie-breaks after drawn games? 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?
For at least some of us, the sort of scheme you propose is an unsatisfactory way to break a tie/settle a draw. If the tournament is being played under classical time controls, but then every draw is turned into a win/loss by having the opponents play a blitz game, the nature of the game is different. While a patzer like me is never going to defeat a grandmaster (or even national master) at any time control, different players excel at different types of events, and it's not fair to someone who's good at classic time control chess to suddenly decide the winner of a game via rapid chess. As an observer, I might want to watch a rapid time control tournament sometimes, but when I'm following a classic time control event, I don't want to see it decided in a short time control tie breaker. (Yes, I know other people will have differing opinions.)

Also, why is a draw so bothersome? Is it necessary for every game to have a winner for you? Personally, I'm accustomed to the nature of chess involving lots of draws at the highest levels. Admittedly, I don't like pre-arranged and quick draws, but there other ways to handle that.

Of course, for a tournament, eliminating draws does not eliminate ties (although it may make them less frequent) at the end of events. While players' final scores may be more scattered, it will still be possible for two or more players to finish with the same total.

Addressing something you said in the last paragraph: While it was a match, not a tournament game, a famous example of using rapid games to decide a winner was the 2016 World Championship, which was decided by a series of rapid games after being tied at the end of the scheduled 12 games.

As for comments from top players, the Wikipedia article linked to above notes that former World Champion Anatoly Karpov didn't like using rapid games to decide the (classical) World Champion. Former candidate Yasser Seirawan also disliked it. Carlsen himself expressed a desire for a different format, but it wasn't clear whether he dislikes the rapid tie breaker.