I know that feeling. And the calling as soon as the surgery opens, and the line is busy so you just keep hitting redial.... and if you're lucky you finally get through. It all adds so much extra stress to booking a doctor's appointment, the whole process makes me very anxious.
One surgery I used to be with had online booking, 24/7 - you could see who was available, compare dates and make an appointment any time of the day or night. I loved it. But my current practice doesn't have that. They do have an excellent reception team though, they have a roomful of people just answering phones - not the same people on the front desk checking people in.