I’ve found a solution for this problem:
The solution to not restart the mission when the instructor asks us to turn on the engine is to make SURE that the collective is at 0%.
I was having the same issue, and I noticed my collective was at 0.7%, so I forced my HOTAS in order to ensure it was at 0%.
The engine picked up and the lesson was not restarted. The lesson progressed and I completed it successfully.