Does healthcare insurance pay for that in the US ? In the EU, this is considered a personal life-choice procedure and very rarely handled by insurances - like cosmetic surgery. (being the EU though, you can get it for free using public healthcare)
There may be a confusion here, I'm referring at Private Healthcare and not Public Healthcare run by a private company.
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 ?
The French system is hybrid. There's public and private health insurance, and public and private hospitals.
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.)
Sure, but if you for some reason don't want to give birth at an NHS hospital then your health insurance probably won't cover the costs of a private hospital.
The UK and Spain have a private health sector with Clinics, GP, ... To access those you either need to pay cash or have a private health insurance.
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.