I agree with you that formally speaking, initial states may have no outgoing transitions. We can consider such kind of structures, and there is no good argument why that should be forbidden. The semantics of formulas is defined in a way that copes well with deadend states: All universally quantified formulas Aphi would hold in those states while all existentially quantified formulas Ephi would not hold there. Even A false would hold there, and even E true would not hold there.
In practice, such structures are however questionable and should not occur. In particular, reactive systems must have ongoing computations, and if an initial state would not have any infinite path, then this would mean that the system is somehow broken in an initial state. For this reason, you may also have good reasons to define that initial states must have outgoing transitions, but we did not demand this.
So, the exercise tool should be fixed accordingly.