Whilst I certainly wouldn't hold up the US system as an example (as it's about twice as expensive on percentage of GDP than European models), the British NHS is a complete mess and it really doesn't work very well at all. It's massively bureaucratic, poorly run, and extremely wasteful.
One of the biggest problems is the system gives people no incentive to stay fit and healthy. It's regarded as a free service and as a result is misused by all sorts of timewasters who really need a kick up the arse, not health treatment.
The funding model should be through individually rated public health contributions, not a flat rate on general taxation. The people that don't smoke, don't end up in the casualty dept pissed out of their minds or drugged up on a Friday night, and who maintain a sensible weight, should get a discounted level of contributions. All the idiots who clog up the system due to their behaviour should pay extra. Give everyone an annual health check at the doctors, and set their contribution level for the following year based on how healthy they are. People would soon take the NHS more seriously if the ones who create the burden had to pay twice as much NI contributions than those who take better care of themselves.
But who would decide what each person pays? And where is the money going to come from to pay for all this extra bureaucracy?