Certainly he does, or rather the proprietor of the company with whom your school has the contract for the service.
Properly speaking, the pupils do not come under the school's control until they enter the premises and, while they are travelling to and from school they remain the responsibility of their parents. There is no essential difference here between travelling by a regular public transport service and by a hired vehicle contracted for the purpose: the responsibility for the conduct of passengers rests with the provider of the service.
Not all bus drivers are adept at persuading young people to behave and inevitably, in the absence of parents, they turn to the school for help. Inevitably, too, because schools wish to promote good behaviour and maintain a reputation for so doing, heads and deputies spend an inordinate amount of time dealing with their complaints.
Nevertheless, the responsibility does rest with the contractor and, if there are problems, a non-driving assistant could be provided by him to ensure good conduct and, in particular, that the driver is not distracted