The systems are closer to American than UK.
The premium you pa in USA is based on your health, and lifestyle. THAt often means poorer people paying a higher premiums, which inevitably leqads to reduced cover, for the poorer people.
The German and French model;s are based on your salary, as far as the part collected by the government, which covers about 70% of the cost.
The remaining 30% is covered by private inurance, which isn't compulsory, and is covered by a mix of your employer (if there is one) and your own private cover. If there are noi claims in any one year, you often get a partial refund on your premiums.
It's not free at the point of delivery in either France or Germany.
If you don't pay at the point of delivery it is free "at the point of delivery".
Then even if you do pay "at the point of delivery" it is only a small percentage of the real cost. And many ancilliary services are free or very cheap, e.g. blood tests, xrays, etc. I think it's something like 25€ for an Xray. And you walk out with your Xray to give to your doctor, etc. And another is sent directly to your doctor.
It's about 25€ to see a doctor, which is reimbursed by the private insurance automatically.
If you need a hospital stay, you have choices about the quality of the room e.g. TV, phone, internet, etc/ private room, etc.
Obviosly your premiums are based on those choices.
Oh, and you can usually get a routine appointment face to face within about 2 days.