I can understand that if everyone had games left open indefinitely, it would cause problems. But I doubt many people would do it. And you could always have a limit. Say, 10 that can be open endlessly at a time. After that, other people are warned that due to all the open games, their time will be limited.
But short of that, is it really too much to ask for more time? If not 20 minutes, why not at least give us 15?