A desktop application would be better, as it would be easier to implement a sound notification feature. I don't think it should be enforced, as it would only increase the activity amongst those, who look for games the way Bonhert described (myself included). So it only adds one more layer of possibility how to find a game.
The big thing in my opinion could be that the system would notify you whenever one of your cup opponents is online on the site and/or #AG. That would be nice, just a modal window on the site saying: "Hey you lazy f*ck! Your opponent you didn't bother to play with is available!" or something similar and less harsh perhaps.
But that's getting off the rails a little. About the topic, I was preaching for something like this to implemented years ago, without much success.