It's an imperfect system but way better than what you have, it's just very badly managed.
I recently had a tooth abcsess flair up at 5.30 on a friday. My dentist (private obviously she can't be bothered with the paperwork) was finished for the weekend so I went to the all night clinic at the local hospital.
Three doctors standing three feet from me having a chat.
I asked the receptionist for an appointment with the doctors but told I'd have to ring the PCT 20 miles away.
She handed me the phone

and I told them I just needed some antibiotics.
No you'll have to go to the dental clinic 30 miles away on Saturday morning

.
Soooo what if I had an abcsess on my leg, could the doctor here see me - yes - ok - I've got an abcsess on my leg.
Can I have an appointment? - I'll call you back
90 minutes later the receptionist rang them and said look the doctors will see him.
Got in to see the doctor who had heard the whole thing he said "so you have a tooth abscess here's some antibiotics."
1 other patient had been in and three doctors who had been shipped in from abroad had just been sitting around +1 receptionist + 1 person in the PCT.
It didn't use to be like this. In France recently we needed the doctor, went to the local evening surgery queued for 30 mins and saw the doctor, that's what we used to have.