Well, I've sometimes also tried castling queenside as Black in the French C01 (in order to generate an asymmetric & less drawish game). The problem is the open e-File, neither White nor Black can ignore it, and if they do and instead go for a pawn storm towards the opponent's king, then they're likely to learn the lesson: Don't attack on the wing when your opponent can play in the center (e.g. double or triple their heavy pieces on an open e-File).
I've seen several master games where the dominance on the open e-File was more decisive than the attempt of one of the players to make a pawn storm on the opponent's king. So basically also with castling to opposite sides of the board the players will need to play their major pieces to the e-File (if they want to avoid a desaster), everything will get exchanged and again you end up in a very drawish endgame.
By the way, this is also your very best defence as Black against White players who think they can afford to make a pawn storm on your king:
Play for the dominance in the e-File and if they keep on ignoring this and instead keep on advancing their pawns towards your king, they will end up losing.
Btw. the same seems to be true for those variations where either White or Black wanna unbalance the game by accepting an IQP in return for some active play. Due to the open e-File the position can be simplified so quickly to a bad endgame for the player with the IQP, that it's actually very risky to play these early c4 (White) or c5 (Black) plans in the French Exchange Variation. And again, if the player with the IQP doesn't wish that the position gets simplified and chooses not to oppose their rooks (which means they grant their opponent the control), then they'll learn the lesson, that playing asymmetric plans in a symmetric position with an open file is antipositional. The open file dictates the correct plan, and is by far the most important positional feature of the French Exchange, such that none of the two players can afford to ignore it.
And hence French C01 remains drawish.
Nevertheless there are of course good winning chances for both sides, because with lots of experience you'll start to notice the 'minor' inaccuracies of your opponent, and if you manage to accumulate these minor advantages, then you may end up in a winning endgame. It's a great variation to practice accurate positional play. Several friends in my chess club have sometimes told me I should abandon the French Defence (I've been using it for 17 years) in favour of Siclian, because they say that as Black I can't play so well for a win with the French due to the drawish Exchange Variation. I don't believe them:) Actually, having an equal game only after 3-4 moves is more like a great achievement for Black, which they never have so quickly in the Siclian. And once you have equal chances, it's only a smaller step for you to get the better prospects (e.g. due to inaccurate or also anti-positional play by White).
So IMO the Exchange Variation is not reason enough for me to abandon the French in favour of the Sicilian, French is not worse than Sicilian if Black hopes to play for a full point. Maybe on a master level this is true, but on our amateurish level of play < 2200 Black can equally well play for a win in the French Exchange Variation, as they can with some super-sharp Najdorf. Only my opinion, can be totally wrong.
I think this is my favorite way to play, castling opposite and flinging the pawns forward, works suprizingly well.
What does black play vs the early c4 plans if he is trying to adopt this setup?