I came across your question while trying to figure out how I can set individual start points for each show, but I don't think that's a feature. I am constantly fiddling with playouts and schedules and would absolutely love to know how to set individual starting points for shows (if that's even possible).
I did however find that I could start the schedule at a random start point by selecting "random start point" when creating a schedule. Did you happen to pick this for your schedule?