This is the problem with socialized medicine. You don't really have the option and you're stuck with whatever doctor the system gives you.
There is no particular reason both have to be implemented at the same time. Maybe its culturally popular but there's certainly no technical reason.
I live in a former 1st world non-socialized medicine country and I have no option in where I go for medical care other than on a very meta level of which employer I work at and where I live ... in an emergency (heart attack, car accident) the ambulance drives me to the closest ER, there is no free market deciding which is "better". In a non-emergency, my cheap health insurance contracts with precisely one large medical chain in my area, so if I'm willing to do something like medical tourism and fly to another state I could select a new doc, but again no free market ...
This is a painfully accurate description of the USA
I can change my GP at any time, and I do not have to give a reason.
When I go to my GP for a problem and she thinks I need a referral I then get a choice of 4 different doctors and I can book it online.
You can always ask for second opinions. It's simply untrue to suggest that people don't have choice of doctors in England.
It's not about lack of choice. The choice represents the doctors' competition with one-another. Obviously, as I've pointed out, the choice also has to have meaningful repercussions for the doctor involved, otherwise it's not really the kind of choice that can make a difference.
Many people/doctors may be bastards; many people/doctors may be knackered; many people/doctors may only be in it for the money. But they're still the minority.
I can think of 1 dentist experience and no doctor experiences where I've felt that the person had no interest in their job.
They get paid for patients on their list; they attract funding for the surgery performed; etc.
Ironically, often in the US I've had less choice in providers due to network constraints on insurance.