For what it's worth:
In France, once a "self-sufficient" has been there three months, they can qualify for PUMA. Private Insurance in France is typically a "top up" cover, from what I can see, although you can still get CSI. The cost to join PUMA is about 8% of your taxable income above around 10,000 euros a year.
The change for self sufficients and students (and all their family members) to have Comprehensicve Sickness Insurance, was added to the 2004 Directive at the request of the EEA countries. This, like many other EU rules, then allows that EEA country to choose what they will give and what they won't.
That also allows for future governments of that EEA country to change that again and if they wish, to apply any changes retrospectively if it is in the country's favour to do so. e.g. One country might give citzenship quicker than others; one may give use of their health system; one may allow unmarried non-EEA citizen partners; one may give full benefits when they arrive etc, but then any future governent can end that and apply that change retrospectively.
The UK only started to bring in the requirement for a Comprehensive Sickness Policy, for some of these, in 2011 and now it is everyone. You can see those changes on the Advisories board on here. But as the UK law still said that all those lawfully in the UK could have bill free use or the NHS, the UK still had to provide that. With the new law now ending this for the NHS, it will be up to future governments to decdide if they will still continue that, and that is whether the UK stays in the EU or not.
As the EU rules state that EEA workers can only have what a citizen of that country can have, any changes to that also affect these EEA citizens/all their non-EEA family members/ those using European Court of Justice Rulings to be in that EEA country. We have already seen EEA goverments in the popular countries for Free Movement, do this. e.g. the Netherlands bringing in a flat rate of Housing Benefits for the whole of the country; various UK governments changing the rules for what the UK will pay for with healthcare and new Welfare Reform laws.
Or these EEA governments put through new laws to strike out what the ECJ has ruled they must give to those using Free Movement when EEA citizens have complained to the ECJ, or that EEA country will ignore what the ECJ/EU has said..