There are different types of insurance. There is a low deductible insurance which is more expensive but you pay very little excess nothings like doctors visits. My wife is exciting and she is on this type of medical insurance - the entire process from initial check ups to birth, to follow ups will cost her around $50 out of pocket.
There are also high deductible health insurance plans that are cheaper like mine. I have in theory $2000 per year deductible, I.e. any medical expenses beyond preventive care and check ups I would pay until the $2000 mark. However, as is typically the case with a high deductible plan my employer pays me $2000 a year into a high deductible health savings account (HSA). HSA contributions are tax free and can be used to cover any excess/deductibles, and any outstanding medical costs, as well as things like contact lens solution, condoms, tooth paste.
The end result is I don't pay a penny and I have thousands of dollars in a savings account I can use for anything vaguely medical. The savings account will belong to me even after I leave the company.
A vast majority of Americans have access to better health care than the NHS for far less personal cost. The real issue, and it is a big one, is the poor and long term unemployed that aren't provided with a company health insurance. Even then, Medicare and Medicaid covers health insurance for elderly, disabled, destitute etc. The issue is those people that earn just enough above the poverty line such that they aren't eligible for state social care but their employer doesn't provide them with insurance.