Suppose the last moment the car was at rest is at $t_0=0$. What time is the first moment the car moves?
Suppose you pick some number $t_{first} > 0$. It doesn't matter what number you pick. You can always show that there is a problem with it.
There is a number $t_{first}/2$. This number is bigger than $0$, so it is after $t_0$. So the car is not at rest. It is before $t_{first}$, so the car has not started moving yet.
This is a contradiction. It shows you cannot pick a moment after $t_0$ that is the moment the car first moves. You must conclude the car starts moving at $t_0$. So you are right to be suspicious of his answer.