Suppose a head coach designates her assistant to be the head coach during parts of a game. But only one is coaching at a time. What is the proper ruling?
Is this allowed? Your interpretation is correct. They need to decide who the head coach is before the game starts and that decision is good for the entire game, no flip flopping.
I am not sure where the officials got the idea you could rotate the position during a game. With that logic, you could end up with 4-5 different head coaches with as many assistants as some of these teams have. In fact, a few teams have co-head coaches as appointed by the school board.
They wanted to rotate quarters which is not permissible. They must alternate games being head coach.