What makes you think they are "washing their hands of responsibility" though? That's the interpretation that's baffling. They are doing what the school board is supposed to do, what government is supposed to do, and what they were elected to do.
Parents hand their kids off to the school in the morning. They get them back in the afternoon. Between those times, the school - the teachers and principal - have responsibility for what happens to the students.
If the prayer time has nothing to do with the school, then the school is not supervising and caring for the students, as is their responsibility during class hours.
It's the duty of the school, principal and teachers to adequately supervise and protect the students under their care between certain appointed times. If the school, principals, and teachers have nothing to do with the prayer service, then there is a "hole" in the care and supervision they have a duty to provide.
Consider a hypothetical situation: two of the students get into a fight during the prayer service in the cafeteria. What would happen?
If the director's statement is correct and the prayer service really isn't a school activity, then
nothing would happen. The fact that the students got into a fight would be no concern of the school. In their eyes, it would be no different than if the fight had happened at home on the weekend.
However, I suspect that if this actually happened, the school would become involved and the students who fought would be punished by the school... indicating that the school is actually involved on some level with what goes on during the prayer service.
Edit:
The school board is the level of government that is supposed to work directly with communities in regards to the students of those communities.
The school board is also
the level of government that's supposed to supervise schoolchildren during school hours. The director's statement implies that they're not doing that during prayer time, despite the fact that it involves schoolchildren and it occurs during school hours.