jollysanters from the wiki here it looks to be a similar system to france with universal coverage paid for via a compulsory system, with insurers unable to differentiate on price due to age, health etc and not able to decline. The true private element looks to be the optional top-up coverage.
However you want to cut it, the "private" systems in countries like switzerland & france are not actually private at all, but a method of administering and paying for a system which delivers the same goal as the NHS ie universal healthcare.
The fact is that the WHO put France (which sounds like a similar system) in at no 1 in the world when they did their study. However the costs for us to switch to that would need to be very carefully considered before changing wholesale in the UK. Is there really a need. Not sure.
I also think that your method of comparing costs is not quite right. You assume that because you paid more in total tax and NI in the UK, that a lot of that contribution was for healthcare, mentioning only unemployment and disability benefits otherwise. The UK and Switzerland are very different countries though. Switzerland has a tiny population compared to the UK, and I would guess a quite different population demographically, on various measures. It has a very different approach to tax in general I would imagine (being the world's most famous tax haven) and also think about something like the armed forces, a quick wiki tells me that UK armed forces cost us 2.5% of our GDP, while Switzerland spends 0.9% of it's GDP. I think that the situation is probably a bit more involved than concluding that due to your relative tax situation, the cost of health for you and your entire family in Switzerland is more than it was to you in the UK as a single person. Most of the money we pay in taxes goes to paying old age pensions, AFAIK, did you factor that in? That sort of thing. Do you see what I mean. I have no doubt that the swiss system is perfectly reasonable, as are the ones in the UK, France, Germany etc based as they all are on an idea of a universal healthcare system.
Overall I think that ideas to improve the NHS should of course be welcome, but many on this thread have said "what do america do, let's do that" when that is the MOST expensive system in the world and the results are terrible. And the idea that people in dire need of help should be "turned away" if they, what, look a bit forrin (people haven't really explained how that should work) is just really distressing.