I doubt the assistant cares. School coach of the year is an equivalent of an age group award at a small 5K.
I'm in a similar situation as your assistant. I do all the workouts and try to get and keep kids on the team.
The head coach deals with paperwork and the administration and parents and meet directors and all the other BS. She gets paid for her position, I don't. There is no way I would switch roles with her for her paycheck. She does the hard part, I do the fun part.
If my school had an award like this, she'd have won it already. I wouldn't care. Everyone knows who the head coach is, outside of the athletes and a few parents, no one really knows me or any of the other assistants. If she won an award, it would be an honor for the whole coaching staff.