Personally (as a British citizen paying British taxes and paying taxes on over £50k) I don't mind it much. I've been to the hospital once in my life (when I was born) and I've only been to a doctor once in my life too (I was 16, given antibiotics) and I would much rather I personally "lose" money on taxation while everyone in the country gets the healthcare they need than an American style system where I save money but there are people that can't get the healthcare they need. I also have private health insurance.
As a private individual that does not make use of national services National Insurance isn't great but as a British citizen it's fantastic. My other tax paying family members feel the same.
I agree that it's a fundamental property of modern societies to ensure universal health care, one way or the other. What angers me about the UK model is how the system handles a scandals like Stafford Hospital: with anything less than total fury. At Stafford Hospital (est. 400-1200 patients died as a result of poor care 2005-2009). One manager resigned, one was suspended on full pay. Years later, some meagre compensations has been paid out, some apologies and an inquiry which one party will now use to push the same reforms (maybe they're sensible, maybe they're not) they always wanted to push, and the other party will try to block them for the same reasons they will always try to block them.
Yadda yadda yadda, in the meantime, people died, no-one felt any consequences. The same people in the same organisation that let this happen are still here. And no, I'm not vengeful-blood-in-the-street-type person, but I'm not seeing the whole "Shit, this is a systemic failure, we're all part of the system, thus we're all responsible. How do we fix this?"
£50,000 Employee: Tax £9,822.00 NI £4,214.64
£50,000 Self Employed: Tax £9,822.00 NI £3,217.03
There's no way to avoid that, unless you're willing to accept that poor people get no medical treatments at all.