In its current form, the NHS is in danger of becoming an insoluble problem
You've noted the two key parts of the NHS and both are in big need of reform in my view. We are currently trying to provide a perfect unrationed private system for free and it just doesn't work. Health demand will always be huge so needs rationing. This needs to be tackled through incentives to consumers of healthcare as well as a more honest rationing of the service that can be provided. As for delivery, a mixed economy and attempts at genuine competition between state and private feels like a good place to aim for.
Simon, without some fundamental re-purposing and re-design, the NHS will surely continue to fail. It is the scale and scope of the change that makes it a nettle that few, if any, politicians will wish to grasp.