All well and good...
So having found this loophole has it been closed?
In order to book you either need your NHS number or Name (as registered at the GP), DOB and Post Code.
If you have already booked appointments then you cannot proceed without the reference number so unless I am mistaken you only know if someone has not had a vaccine booked through this service.
Many people will have received letters with an appointment or letter with a phone call. Are these also in the same system?