If the Cowboys keep him on the final 53, he'll be an emergency backup, or somebody to play in mop up duty in blowout losses or wins (though when was the last time we've ever seen that?). And only because there's absolutely no one else available either on the Cowboys or somebody cut by another team.
Reminds me (not in a good way) about 15 years ago, one of my sons was moving up to the next level in youth baseball, after tryouts I was called by the league head and he asked me if I would coach my son's team. I replied that I never played much baseball, knew a little because I'd been around while my two sons played, but that I was in no way capable of being a good baseball coach. He called back an hour later and said that there was absolutely nobody else who could do it, and that I'd get a couple of seasoned coaches as assistants to helpe me. so I took it. But our team wasn't very good, my assistants weren't at practices due to their jobs, we won very few games, though somehow we did manage to hand the league champion their only loss of the season. Caught a lot of grief from the other parents about the team's poor play, but I did my best.
If Green stays I won't give him any grief if I can help it, he will play the best he can, who wouldn't stay and make a few hundred thousand dollars for a few weeks' work?