Have to say, threads like this make me very grateful for our GP practice.
Telephone appointments: can always get through and then there's semi-triage from
the receptionist but if it's urgent you get a same day call back from a doctor. If it's non-urgent and you want to speak to "your" doctor (because they know your history) you can make an appointment for them to call you, but it might be a few weeks hence. The doctor will then either prescribe/advise/refer from the telephone appointment or arrange a face-to-face appointment.
This is inner city Glasgow.
My dad's surgery, other side of the city in the suburbs, is the same.
Ds, a student up in Aberdeen, woke up in the morning with a stomach ache (this was in the middle of one of the Lockdowns) and contacted us about 8.30 saying he wasn't feeling well. We eventually persuaded him (via WhatsApp) to ring his surgery which he did about 11. Got a call back from the doctor about 3.30. She assessed that she needed to see him immediately and he got to the surgery at just after 4 (a friend gave him a lift, despite Covid). By 5pm he was at Foresterhill (the big hospital in Aberdeen) and by midnight he was going in to theatre to get his appendix taken out which is what I suspected as soon as he described his symptoms 
My dad got a pacemaker fitted on the same day as seeing the cardiologist (on another matter but he a retired radiologist himself had commented that he was experiencing an irregular heartbeat, which he could see on his Apple Watch)
(dh and I had to play musical cars to get his car home from the hospital as he wasn't allowed to drive home to get his pyjamas etc as he could have fainted at any time
).
So the NHS can work well 
None of which helps IamaDancer
- but I just wanted to illustrate with my family's experience that the NHS is not dreadful everywhere.
I can only suggest that, in addition to ringing daily trying to get through that you register an official complaint (or "comment"/feedback) about the lack of accessibility. If you have to do that by post because of the lack of an on-line means of contacting them, then so be it.