You seem to be overlooking the fact that, in the UK, there is a private system operating alongside the NHS. This enables the wealthy to choose when to have elective surgery to suit their busy schedules, and ensures that, for them, urgent issues are dealt with immediately.

As someone else has said, the real issue is human resources. In the public system there are not enough practitioners to meet the demand.

Whatever the method of paying/charging, there will always be differences of availability of service between the suburbs and rural areas.

Ireland has something like the Bismarck system you describe. Most health and social care services are under the control of the state Health Service Executive and this body is the main employer. Because of the nation’s history some hospitals remain in the ownership of religious trusts although run by the HSE.

Children, the eldelry and the cronically sick (eg cancer pateints) get free treatment, everyone else has to pay (via private insurance) although such payments are tax deductable (meaning the net cost is reduced by the amount of income tax that would, otherwise, have been levied, assuming that one’s earnings are above the tax threshold.).

We still have a waiting list problem for elective surgery and people awaiting treatment on trolleys in corridors. Unlike in the UK, we don’t have to wait days for a GP appointment.