On your second point, I'm not sure, but I think it's general (ie not only banking) that an in-house scheme is either optional or compulsory depending on the deal the employer has negotiated with the insurer; if it's a "whole-house" deal (with subsequent savings for the employer), it is legally binding on all employees, you can't opt out of it. If not, you can, but it costs more.

On the first, that would be a big change you're suggesting. I don't know how well it would work. An example is Medicaid in the States, but there are numerous problems there, particularly with low-wage earners who are not "poor" enough to qualify but don't get a scheme with their jobs, and aren't rich enough to fund their own.

But essentially, you seem to me to think that a private system (bar a kind of Medicaid) would be more efficient than the public health service à la française. Where do you think that public service goes wrong?