I agree with Gagajo that the current system does not work for anyone other than those who don't work, or who have somewhere private to take calls when they get there.
We have what sounds like the same system. You have to ring in at 8.00am and keep pressing redial until you get through. this can take a while, and if you fail to get through you have to repeat the process the next day. Do you take a day off work at this point? You really don't know.
Also, when I was working, I was commuting at 8.00. The very act of making the call meant that I was going to be late. If I got through, I would be allocated a call, but not a timeslot. So if my call was considered urgent, the call could come whilst I was travelling to work. Are you supposed to discuss your medical matters on a crowded bus? When I got to work I was lucky in that I had my own office, but what if I had meetings, was giving lectures, whatever? Many people have nowhere to take calls, and a lot are not allowed to have their phones on in work time.
Even for people who don't work, the system is flawed. How do you keep a room private in a shared house, or answer a call about something private if your kids are under your feet? What if you have an abusive partner who can hear your calls? What if you want to discuss contraception, and your domineering father is in the room?
Finally, how does a GP diagnose something like arthritis, or an allergic reaction, or anything that requires a physical examination? They have to ask you to come in, so by now you may have lost a day to the first attempt, been late for work, perhaps missed the callback because of working, had to make elaborate arrangements to speak privately, (or maybe give up on the call if you felt threatened) and you still need to go in for the appointment that you once would have had in the first place.
It is reasonably ok for me these days. I don't do a lot of work, and what I do can be arranged to suit me, and I don't need to hide anything from my husband, but I am well aware that this is not the case for most people, and also that if people put off going to see the GP for the above reasons, they are far more likely to end up needing treatment somewhere, and going to A&E. Or worse.