I'm sure teams would try to game the system (a guy playing 5 minutes instead of sitting out), but just have a progressively more extensive fining system (with a shot at losing 1st rounders for real bad offenders) and I think that gets around it.
Actually might be tough to game the system with how much players (and award voters) care about averages. I don't see many star players agreeing to go out there for 5 minutes and tank their averages.
I would extend your proposed rule to the team level: only 1 player can use his free rest game at a time. If you want to rest 2+ players, only 1 qualifies to use his free game off, the others have to sit for the 3 games. That would hopefully stop teams from resting all their starters on the same night.
edit - I think the complexity comes with DNP-CD's. Aaron Nesmith shouldn't have to sit out 3 games every time Ime doesn't play him. But if Tatum, Brown, Horford, etc. are in uniform and get a DNP-CD, clearly taking a rest day, how do you police that?