My understanding of it is that an aerodrome with an approach, holding or let down aid shall have an approach control service (and hence licenced controllers) at the periods/times for which the aid is notified as being in service. So I suppose it is possible to have an unlicenced aerodrome that has an approach aid and licenced ATCOs.
However, I also thought that the public transport of passengers had to take place from a licenced aerodrome (with certain conditions).
ryl