There is just no limit to the demand on the healthcare service free at the point of use. There is no link between demand and costs, the only control is making people wait for treatment.
The alternative doesn't have to be the US model of pay as you go, with some people being bankrupted by medical bills and others having world class treatment on tap.
Other systems (Germany, France, Australia) all provide excellent healthcare provision, funded by a mixture of employers/employees with family provision for children and country wide provision for pensioners.
People can use private insurance, or just pay as they go. This option tends only to be picked up by people temporarily working in the country who will move to other countries before becoming pensioners. Importantly, private healthcare is not significantly better than the standard provision.
The key difference is whether the system is organised for profit as in the US or nit as in Europe.
Overall, It does cost a bit more per person but then there are no significant queues so cost per procedure is probably no higher than in the UK ( where so many people have to queue so do not cost anything while they wait. The type of procedure they make need can be more complicated but that doesn't show in the stats ).
There are also no tricks like paying for an initial consultation privately and then being fast tracked into NHS provision. No one likes to think about the people who therefore stay in the queue for that much longer before seeing the consultant and getting into the system for NHS treatment.
I'm a great admirer of the NHS but really, nurses and doctors and everyone else should not be working overtime, starved of resources. It really doesn't have to be like that.
It's not unlike schools, if everyone has the option of a place at a hood, local school, there is much less pressure to go private.