As there is no stupid question....(this one might qualify..);
Why let the student start up on their own?
Is it wise as an instructor to board an airplane with the engine running?
What example are we giving here?
The flight is either booked out as a solo flight in which case the student is responsible; or the flight is booked out as a dual flight during which the instructor is responsible.
I would not allow a student ( no matter how experienced) to start and taxi unsupervised if it is to be a dual flight.
I might be missing the point here, but how much instructor time is saved by waiting inside till the engine is running?
I would hold the instructor liable for the damage, and since he works for the school the school will have to swallow the loss.