Yes it’s true, the role of a Producer is really to make sure the production of the game is on track while assisting in Quality Assurance. Bottom line is that having a Producer who is not a hardcore gamer, improves upon the chance that the game will be playable by casual gamers. If the casual Producer does his/her job the game should be much easier to play and accessible to all then if a hardcore Producer oversaw the project. This inreturn means the game can reach more market share AKA more money.
Umm… I disagree!
Producers most definitely need to relax & play games, and forget schedules & plans sometimes 
Producers need to play their game and make sure that everything is good, that is part of their job. Other then that they use a stress ball to relax them as they create their tables and graphs.