In my surgery they have a set number of appointments that gave to be booked on the day, by telephone at 08:30 hrs.
They usually go by 08:45 hrs!
However, if enough fuss is caused then miraculously appointments are suddenly found. In reality they keep back a number of apps incase, like you patient's need to see a Dr, provided you get passed the receptionist.
Well done for giving grief but it shouldn't be needed