I think there is a real problem with the perception that private healthcare can step in and fix things as a viable alternative to the NHS.
There are not enough clinicians in this country. If the NHS was abolished tomorrow that wouldn't change, the pool of available clinicians (including the irresponsible, callous, rude, inept ones etc) would remain exactly the same.
The discrepancy between demand to see a given clinician or service and the capacity / availability of that clinician or service would remain more or less exactly the same. Waiting times on average, would remain exactly the same. Provision of emergency services, exactly the same. Additional qualified staff would not spring up from the earth the moment private finance was available to pay (presumably more) for them.
And that would be the case for the foreseeable long term. The difference would be solely as to the balance of funding from the public purse, insurers and private pockets.
Regardless of how you feel about public vs private, healthcare provision is in a state of precipitous decline with no quick solutions available.
The same is broadly true in many other nations because of the inflationary nature of social and healthcare costs. To address this adequately required long term planning and funding provision, not only within healthcare itself but much more broadly in public services and social care - which hasn't happened. This needed to start decades ago, and instead the reverse has happened (a shrinking of the social state, collapse of social care, rise in inequality, and long term attrition of qualified staff whilst demand has continued to rise).
Even the best government right now would have a very hard time fixing where we are, even over the course of decades. Attempts at top down reorganisation of the NHS to address it's many failures and inefficiencies have largely made things worse.
We're in a right state and I personally do not think a public funding structure is really the root cause nor would moving away from this solve much. I think we would have many of the same problems, of not worse problems with a privately funded healthcare service - only it would additionally cost a lot more per capita and this is bourne out both in 1) per capita comparisons of cost between the NHS and comparable private healthcare models and 2) comparisons of how NHS funded services perform and how much they cost before and after they are handed over to private enterprise (largely much worse and at greater expense - has happened to many NHS services including many ambulance services).