It is not about "should", it is about "is".
In the NBA, players take games off.
There are exception. Wallace never took a game off. That's why he and the Bobcats were better in the regular season then in the playoffs. They played at top level through out the season while other teams were less consistent. Come playoff time, those other teams notched it up another gear. The Bobcats didn't have another gear left. They got creamed.
To bring this back to Green, and what makes it frusterating to watch, is that he has the talent to be a very good player. I'd say a borderline all star level talent. But his unwillingness to step up, and be that guy, is perplexing. If Paul Pierce was here and took the night off, that is one thing, but for Green to do it is another
I'm hopeful that Stevens demands him to be be in attack mode 100% of the time, and if not, sit his butt down. Maybe that will get through to him, because waiting for the light bulb in Greens head to turn on by it self, isn't working.