I don't see how getting out with the engine running so as to allow the student to solo has anything to do with letting them start the engine on a dual training flight before you get in the aircraft.
Dual is dual, solo is solo. When I am conducting dual training I am responsible for the entire flight and therefore think I should personally check the airworthiness of the aircraft and be in the aircraft before it is started. If after the commencement of a dual flight I elect to to send the student solo I have already fulfilled my responsibilities as PIC. I see no danger in a properly trained instructor getting out of a running aircraft and indeed consider it a normal competency for an instructor.
Obviously if you are dispatching a student on a solo exercise then it is up to them to safely start the aircraft, a responsibility they have as a result of being PIC.
Bottom line: I think you are short changing both yourself and the student if you only get aboard after the student has already started the engine