IMO, with a commercial lic on the aircraft you are training in, you would be qualified to start the aircraft yourself already, so there is no question of ability (or shouldnt be!).
I would not allow running changes personally for the reasons already raised.
As for whether you should or not is up to the captain (the instructor in this case), so it is his responsibility in exactly the same way as with a pre-PPL student. If it goes wrong its his problem and the school pays, but he may be more inclined to let you start up before he is onboard as you are less likely to screw it up.
JW