The main difference we have from NHS (if I understand NHS correctly) is that since NHS is single payor, the only way they can control costs since demand is automatically encouraged (it’s “FREE!”) is to tightly control the benefits and options available to the consumer (rationing). Being publicly owned, things that are popular and common will be well-funded (maternity care and tooth aches) while things that are more rare, in a specialty, perhaps more extensive are likely to be poorly funded (types of cancers, rare diseases, etc).
With our private insurance system, people have the freedom to pick and choose what “package” they can buy. Some people want high monthly premiums with no copays or deductibles (anytime you need something, you’ve got it fully covered – this encourages over usage, but they pay for it in their premiums). Other people choose high deductibles (they pay for everything up to a certain amount and then the 3rd payor starts picking up a portion of the tab – this will have cheaper premiums, but if you get caught needing care, you’ll be paying for it up till the deductible). And there’s potentially a plethora of options for people to get insurance that meets their desires (generic coverage vs. name brand, tests covered or not, maternal coverage, flu shots covered or not covered, etc)…
However, states and federal government have increasingly regulated things that insurance plans MUST cover which limits the variability between plans. Also, people in general don’t have much choice because insurance can’t be sold across state lines and it is much cheaper to get insurance through your employer because employers get tax cuts on employee health insurance (so if you lose your job, you lose your healthcare). In a free market of insurance, high premium plans would probably be pushed up in price by its “tragedy of the commons-esque” nature and people would then choose more cost effective means of insuring themselves (most stuff would probably be paid out of pocket with the exception of real expensive emergency care or “Major Medical” plans).
Obama wanted a “public option” in his health care reform, which would’ve probably pushed us towards single payor since the rise in private health insurance would’ve driven more people towards the publicly funded insurance until private insurance was run out of business. Instead, that was left out and we are forced to buy from companies that can absorb the costs of increased regulation at the expense of smaller insurance companies that can’t…. So Congress basically chose more fascism over socialism. Don’t let people tell you Obamacare is socialist. Socialist doesn’t force you buy corporate products. That’s fascist.