They reside in the dressing room of the team that employs them. They presumably report to the Head Coach of the relevant team and are therefore exposed to certain pressures that will invariably prevent them from making an unbiased medical decision (even if that's subconscious).
If he's fit for Old Trafford, who do you drop to bring him in?
Woakes (this loses you a relatively strong batsman).
Archer (obviously no)
Broad (been our most effective bowler in terms of wickets so far)
Leach (you would be reliant on Root to be your spinner).
My vote would be to bring him in for Woakes & bring Liam Dawson in to replace leach (and the batting lost in dropping Woakes).