My numbers were pulled from published government expenditure sources for the US and France on the web.
I did some arithmetic, for example subtracting out 4.6% of GDP government expenditure on health care from the 15.3% total health care expenditure in the US, in order to arrive at the 10.7% that must come from the private economy.
Yours is a good point that there is a gap in French insurance coverage which is borne by the individual or paid for by private insurance. There are some private schools and creches, mostly for religious people or for problem children who are expelled from the public system for rowdy behavior. The most elite schools in France are, of course, public.
Of course when trying to assess the different health care systems qualitatively, it is difficult to find comparisons. Longevity and infant mortality seem the most objective to me.
Likewise when attempting to assess the overall quality and utility of public education fulfilling its primary role to socialise children and impart proper rules of conduct and knowledge into them, it is difficult to measure. To me, murder and violent crime rates seem the most objective number. If young men are properly socialized, they will not kill or attack people. However, I expect that there will be objections to whatever criteria are used.
The thing which disturbs me most about the American approach is the extent to which it is driven by debt. American business and individuals are taxed less, but approximately the same level of social services are offered for most (certainly not all) people in the US economy. France taxes more. Both countries spend in the ballpark on government and services. There is no question that the US taxes less...but then it funds these services that it pays for with interest-bearing debt. The low taxes "juice" the American economy - the power of leverage - but leverage only works until the payments catch up with you. I do not worry about the sustainibility of the French model, although obviously it would be better were employment to be improved by making the workplace and capital availability more supple. But I do worry that the American government itself, and industry, and even foreign trade, produce the higher returns that they do because of this insidious debt leveraging, which eventually creeps up in interest due until you cannot do it anymore...and then there is massive debt with slow growth for a long, long time to pay it.
The reason that the debt-economy works is due to economic growth. As long as the percentage of debt to the GDP remains relatively constant, the economy will not be affected. Currently the value is almost 70% and rising. If the value goes over 100% the economy will become more unstable. If the economy performs poorly (lowering GDP), the debt to GDP ratio will increase providing positive feedback--making the economy perform even worse. For this reason, the US economy needs to be constantly tweaked.
I'm not an economist but I assume that as long as the government keeps a close leash on the public debt to GDP, the economy will operate normally. If the GDP rises rapidly without an increase in government spending, the positive feedback should allow the economy to operate better.
I don't think it is realistic to eliminate public debt as a way of operating a government. Almost every country in the world operates with a public debt, averaging about 50-70%. With that being said, the US public debt situation is unsatisfactory right now (I won't use harsh words like 'dangerous' until it gets to roughly 150%). It is obvious that the percentage of public debt to GDP is going to be rising for the next couple of years. Concrete conservative action must be taken to stabilize it.