Technically, Mayor Pete had little to do with the school board and their problems. As you mentioned, those are elected positions.
That said, one could argue that he had little to do with the private investment that happened in downtown while he was mayor also and he had no problem claiming that as his success, one of the few he can point to in my opinion. If I'm being totally fair to him, I don't think he's responsible for the failures of the school system. Crime is a different story. Credit where it is due though, he did get Brian Boitano to come to the grand opening of a $20 million dollar ice skating project he had built in a downtown park!