When I was an up and coming player at the Orland Park Chess club, matches were commonplace. It was how we resolved disputes, put an end to smacktalk, and occasionally entertained ourselves. I know we were not alone in enjoying this, just turn on the food channel and there is a good chance you could run into Iron Chef, or Bobby Flay's Throwdown which are all about decisive resolution by showing who was better that day.
In the internet age, great numbers of paper tigers have been created throwing words at their screen with nothing to back them up. For instance, earlier today on the IL Chess forum two very strong players, who are both pretty nice guys too were bickering back and forth as they often seem to, and I thought wow it would be so much better to watch them decide this over the board. Sometimes these matches don't end up as you were hoping for as in the case of the classic grudge match between Bill Brock and Sam Sloan (though that is a pretty strange story in its own right).
Maybe players don't realize that they can still play rated matches, or it could just be the demise of the chess clubs, but these are also tremendous training tools. It is after all a good enough method to determine a world champion.
Perhaps it would be interesting to put together a match play tournament which would be a knockout style event. That would actually be a really interesting way to determine the Denker rep.