Chalker, could you by chance add something to the rules, especially for build events, that would allow coaches to compete in any vacant time slots in events? Not to compete with students, but just for fun, and perhaps a cheap trophy. Something like a wooden spoon with the competition written on it or the like; at the competition/E.S.'s discretion... And, if any competitor were to need the timeslot, they should get it, even if there's already a coach scheduled for it.
I specifically mentioned build events because, to my knowledge, they wouldn't be too hard to accommodate more competitors in the vacant timeslots, if there are any. Whereas, study events would involve more grading time, which would detract from the competitors' experience and the timelines of the Awards Ceremony.
I also mention coaches to encourage alumni to help with a Science Olympiad team, even if they've moved and can no longer help with their own. If you don't have the time to help coach, do you really have the time to make something for a competition that's just for fun?
Interesting idea, but I think you are underestimating what event supervisors of build events are doing during 'empty slots'. Most event have a lot of scoresheets and data to process and enter into spreadsheets for final tabulation, which often is done during this time. Even if they have down time, having coaches compete like you describe wouldn't occupy only those empty slots. It would also require additional time to enter their scores (into ANOTHER scoring system,which hopefully doesn't get messed up with the 'real' scoring system), plus then tabulate the results to determine who is the winner. Finally, at most large tournaments, there AREN'T really any empty slots. They are almost fully booked.