The goal is to get around a "forced" turn when a player does not have any ideas. Of course, once a player has taken a turn, s/he cannot be story teller again until all other players have had their turn.So...what's the point? Just to move the game along faster? Because you still have to wait for all the players to play a card. Also, when you play in clockwise order you know exactly how much time you have before it's your turn again, so you can constantly be thinking up good words/stories for your cards. That way, when it's your turn you are ready and (theoretically) no one has to wait. In this variation, though, there's not even the surety of knowing who will go next, so the fast-thinkers will snatch their turns, and then the slow-players will take their turns, and it will all just amount to a new turn order (based on speed, not sitting position). As far as I can tell, nothing changes.
If the variation rewarded fast-thinkers and penalized the slower players, that would be a real variation to the game, though. You could do this thusly:
(1) Deal 5 cards face-down to each player.
(2) Players look at their cards simultaneously. The first player to think of a word/story idea for one of their cards slams their card down on the center of the table. This player becomes the storyteller and AUTOMATICALLY advances one space on the scoring track.
(3) The storyteller tells the other players the word/phrase that corresponds with his chosen card.
(4) The other players race to place their own cards on the table beside the storyteller's card. The FIRST player to place his card on the table advances his token one space on the scoring track. The LAST player to place his card on the table moves his token backward one space on the scoring track.
(5) As per the normal rules, the cards are shuffled and revealed to the players who then (secretly) bid on which card they think belongs to the storyteller.
(6) BONUS: If the first player to place his bidding token correctly guesses the storyteller's card, that player gets an extra point.
Once the storyteller's true card has been revealed, repeat steps 1-6 (dealing 5 new cards to each player with every hand).
This way the game rewards those that think quickly AND guess correctly. You'll notice that you can also gain more points by being a regular player than you can by constantly taking the role of storyteller, but the risk is also greater.
I'm not saying I like this variant (I personally think Dixit is great as it is and I wouldn't add anything this complicated to a game that's meant to be a fun/social/party game), but if you're (a) trying to spice things up, (b) wanting to speed the gameplay up a bit, and (c) like adding some screwage to your games... well, this would do it.