want what you're missing is that healthcare costs more in America because its private. The American government spend a higher percentage of their GDP on healthcare than we do, because the costs are so massively inflated by the involvement of profit making companies, but they don't have the same level of income from the public to offset those costs as we do.
Have you seen Michael Moore's Sicko? Regardless of your opinions on him, the fact that everyone featured in that film - the ones who are homeless due to bankruptcy, the ones who've died because of insurance companies trying to get out of paying, the ones dead because insurance companies have found ways not to fund treatment - are insured. It doesn't even begin to cover those who have no insurance.
When I was in New York I overheard a shop worker telling a colleague she really had to go to the hospital. I asked if she was okay and she said she had burnt herself a week previously - she pulled up her sleeve and she had a severe untreated burn covering her forearm. It was truly horrific. We got chatting and she told me she would have to get another credit card to pay for it, and she already had lots of debt because her child has severe asthma.
So yeah, forgive me if I'm not overjoyed about the idea of PFIs taking over more and more of the NHS. We are heading for a very scary future where only those with means will be able to access the sort of timely healthcare you're talking about, and the rest will be lucky to access any healthcare at all.