What determines the transmitting frequency privileges of an amateur station?

The transmitting privileges of an amateur radio station are always determined by the license class of the person acting as the control operator. The control operator is the individual in active charge of the station's transmissions. Even if the station is owned by an Amateur Extra class licensee, if a Technician class operator is at the controls, the station is limited strictly to Technician class privileges. The FCC rules focus on the person actually operating the equipment, not who owns it or who else happens to be in the room.