"you're effectively operating under your instructor's FRTOL licence."
No! The student pilot is exempt from the requirement to hold a FRTOL under Article 21 (2) (a) (ii) as a person being trained in an aircraft registered in the United Kingdom to perform duties as a member of the flight crew of an aircraft.
The FRTOL is not a mandatory requirement for a PPL however, PPL training does include use of the radio and being able to operate the radio is an essential part of the JAA Skill Test. If you have got to learn how to use it, including passing the written exam you may as well complete the FRTOL qualification at the same time.