The NHS isn't as good as it could be, and it isn't as good as it should be. But it has to be worth persevering with it.
"Private companies" isn't the answer. They still have to be paid by someone, and that means private insurance. But you can't just convert nationwide fully funded medical services to private insurance. As someone has already pointed out, some people will be uninsurable given their current age and state of health. Medical insurance premiums are calculated on the basis that the proposer is healthy and that all existing medical conditions have been declared (which effectively means that cover for them is excluded or that premiums are loaded). They are related to age, so anyone over 55 will have to pay a very high premium, and anyone over 75 is likely to be declined.
So what happens to the uninsured? Probably the government will have to pick up the tab. Insurance companies will love that - what an opportunity to inflate their premiums! Buy insurance shares now.
As to the management/clinical staffing issue, there's no easy solution there either. I'm sorry to say that the consultants working in the NHS these days are not in any way trained to take on the management functions that are necessary to run any part of the sort of health service we need. They are (mostly) capable clinicians, but even restricting the argument to that area only, they are too often lacking in the requisite people skills.
Of the managers and administrators, to put it politely, one sometimes wonders whether they have the skills needed. We get cross about the banks paying extortionate sumes of money to their managers, but the "pay peanuts, get monkeys" rule applies across all sectors. Not that NHS managers are paid peanuts - they are quite well rewarded these days, but not to the extent necessary to attract first rate people. I feel that NHS HR people and the various selection committees are a significant part of the problem. Too much PC and office politics, resulting in the appointment of people who are utterly incapable of doing their jobs.

And don't even mention the clowns responsible for the enterprise IT plans.

As an elderly, obese, deaf diabetic with asthma, high blood pressure and high cholesterol, not to mention another half dozen minor chronic complaints, I have had to spend quite some time in the hands of the NHS. It ain't as bad as it's painted, but, by heaven, there's plenty of room for improvement.