Sadly, I think there are more people that would be in favour of a private system than we'd like to admit. A lot of people have bought into the Tory idea that the NHS is unsustainable, and that the reason we're all poor is because we're paying for what they see as sub-par care.
On one side, the NHS is arguably the greatest success story of the UK, and I think many people would riot if their free healthcare was taken away. On the other, people will happily vote against a party that is looking to increase its funding, and will happily vote for a party that has made significant moves to privatise our healthcare system, so logically there must be people that aren't in favour of the NHS.
Sadly what most of those people don't realise is that the majority of their private healthcare will take place thanks to the NHS. Going private gets you into the nicer wards at many of the same hospitals, to be treated by the same doctors and nurses, and without any significant delays.
It's all well and good getting your hernia sorted out faster than on the NHS, or getting knee ligaments rebuilt without a 12 month wait for an operation but there's almost no concept of a "private" Accident & Emergency department.
Yep, you're essentially paying for the administration costs by going private, but that's not what the Bupa adverts sell you, and it's not what people seem to assume when you say you've got private healthcare. They think you're in some kind of special institution where you're pampered 24/7 by medical professionals, when in reality you're in the same NHS bed as the person next to you with some minor benefits. Additionally, as you've rightly pointed out, your private insurance will only cover non-existing illnesses. A&E isn't covered, nor is anything that might've existed before you picked up your insurance, and they'll be sure to check up on that. Take away the NHS, and the infrastructure becomes fully private, and that's when the costs will go through the roof.
For the life of me, I don't know why this isn't what Labour are driving home to people. They should be telling people "Vote Labour, or kiss the NHS goodbye".
Best value for money health service in the world absolutely no contest. Free healthcare for all no questions asked. Sounds like some johnny foreigner to me. Probably a Trump supporter chiming in with his alternative facts. Even he can get treatment here. I'm not so sure about a cure.