As far as I know, there are chess tournaments, and, moreover, for more than a dozen years, and all over the world. Prize money is traditionally received by those who reach the top positions.

What you are describing is more like a duel where the winner takes the all bet. Also, I don't think such websites will become popular because the community of chess players is relatively small compared to other games such as card games. Mastering the cardgames like poker for the average player is easier and faster than chess, and this is a fact.