Spain sometimes subcontract public healthcare to private hospital. However from a patient point of view, you are still treated as a public healthcare patient (i.e. you pay through your tax and the government has a specific financial agreement with the company running the hospital).

Otherwise I'm confused by France model. What's the business model of a Private Hospital is they receive no money neither from the Government nor insurances nor their patient ?

For everyday care the public health insurance reimburses 70% and the private health insurance the remainder.

For more serious procedures (such as a baby delivery) the public health insurance reimburses 100% of the fees. The prices are fixed by law, the practitioner can charge more (if it's a private hospital) but then the private health insurance will have to pay the difference.

The public health insurance is paid through taxes (I think something like 5 to 10% of the salary?) and the private health insurance is between 50 to 100e depending on your family and coverage (glasses, teeth, etc.)

I think we'd have been given a shared room for free, but at the time it seemed like a good choice to make.

Both the UK and Spain have also a public health sector, with Clinics, GP, ... that is accessible for free.

In the UK and Spain, very few private health insurances covers Birth or Emergency Care: for those you are supposed to go to the public service or pay cash in a private clinic.

Other countries like Belgium have health insurances that give you the kind of extra comfort you mention. But they cost something like 5 EUR a month vs the 100 EUR in Spain and the UK.