If the doctors are independent, then it becomes considerably more difficult to trick the system. The players can't be the ones deciding, the managers can't be the ones deciding, the team doctors can't be the one deciding. Put an independent doctor at each fixture, just like a referee, and figure it out.
If the FA eliminated its arcane, backwards rules regarding "protecting the integrity of the referee" and actually started handing out retroactive yellows for simulation, you could easily punish teams harshly after the fact for breaking the rules. There is no reason to let concussed players back on the pitch.
It's such an easy fix, but nobody wants to fix it.