Don't bother with the NHS, it's a carwreck. See this, for example: https://www.bbc.co.uk/news/health-67239548
Last month, after years of waiting, I got a letter again delaying my assessment for a growth on my neck, for the foreseeable future. It's a good job that I'd gone private a year ago, as it turned out to be extremely advanced, with neural invasion, requiring two rounds of surgery to be removed.
Nationalised healthcare ought to be means-tested to target help at the people who need it, and not delay others who can afford to pay their way. The population 'pyramid' as it stands can't support the health and social care model we're using, and the lack of market incentives means that we don't train enough medics.
I'm hearing bad things about UK's NHS, and while I don't have any firsthand experience with it, I do get the feeling that the French system is going down the same path.
It's true that I don't usually think about going the private route. I'm not 100% sure, but I think you can probably see a private specialist directly. I also don't know how much it would cost, but probably quite a lot, given the prices in the public hospitals, and I don't think it's covered by the insurance, even though I carry insurance above the standard national scheme.
I contacted the local French doctor, who brought us in that day, gave my son an actual examination (the UK GP didn't do this), asked if I was aware that he had an ear infection (I wasn't, the UK doctor didn't check), and revised the course of treatment which resolved the problem quickly.
I was extremely impressed by the French doctor; he was in a different league to the standard in the UK.
I don't think so; the number of medics we train in the UK is mainly controlled by the medical colleges.
What concerns is that where I live, it's my understanding that, with a few exceptions [0], you can't skip a GP and go directly to a specialist. And I'm not convinced that GPs here are any better than in the US or elsewhere. Bonus points for large parts of the country not even having enough GPs for their population.
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[0] You can go see an eye doctor or a dentist without seeing a GP first.