"Injured" player sits out the drive is one of mine from the 2019 LSU bullshit.
Wrinkles to this include visit with Sideline Doc to approve return, and/or sitting out the quarter, or X minutes of game clock. Players would have to leave the field immediately, only emergency treatment on the field and player experiencing the emergency is disqualified from further participation. Limited number of personnel allowed on field, no sideline gathering for coaching - or whatever - while in "injury time", no coaches onto field for the same.
You want to coach 'em up? Call a timeout, Coach.
On the other hand, visiting teams would be allowed to bring their own HVAC repairmen.
Oh, and if a ref is going to stand over the ball and not let the play start, then the play clock stops and either restarts or goes back to 25 when D subs are in. Under this, the game clock doesn't run when the ball is not ready.