I understand what you're saying but think this simply might not work and would cause rifts between players and teams. Are they gonna pay your salary if every team doctor says you're able to play but your guy advises you not to? Who are you going with as a player then? Your own guy probably but then it's free for all after that.
Players checked for concussion should be cleared by independent medics though, although maybe that's already the case? But of course in what sense are you "independent" if you're hired by the NFL to perform such checks, whilst the league has every interest in keeping their star players in the games...