I think the 9% is both publicly funded and privately funded. The same is the case is the US.
Private healthcare could be 30% of GDP for all I care because it isn't paid for via general taxation. If everyone earning £50+k wants to spend £5k a year on healthcare that's not my business.
Neither a low nor high % is good or bad in and of itself. But a model that is predicated on growing by more than 1% per decade of GDP is obviously unsustainable.
Bear in mind the current health budget is 19% of tax spend. So assuming that total spend is correlated to GDP over the next 40 years we will be spending 38% of tax take on health... To afford this we'd need to either privatise the entire education system or privatise everything else apart from health, social protections and education. This is ignoring the fact that social protections are forecasting the same level of deficit, meaning the only two departments that could be financed would be health and social protections. We'd have to eradicate education, defence, general public services services, foreign aid, police etc.
And in terms of "austerity" how Labour have been able to sell this false myth is beyond me. The Tories have been taxing and spending at a greater level than ever before. The country feel like they've been victims of austerity because we've been allocating more and more money to the pyramid scheme departments at the expense of other departments. Show me on the green line below where the alleged "austerity" is?
To show what austerity actually looks like a similar image for Greece (viewing the average line focusing on 2008 to present))