For what it's worth this is quite similar to my experience here in Sweden except:
- you speak to a nurse on the phone, not a non-medical receptionist (frankly, that sounds insane)
- the Vårdcentral (equivalent to surgery) prioritizes children in my experience
- your Vårdcentral is not allocated, but you can choose one. Some have better capacity, some have worse.
- there is another layer between Vårdcentral and Akuten in a hospital called Närakuten where you can go to get checked out. The waiting times are anywhere between 30 minutes and a couple of hours.
I would even guess that the British situation is not unique but the medical system is overloaded everywhere in Europe.. Most people I know, including myself pay for private care.