Index, the answer, if you don't want the NHS system, is to do what the Netherlands do.Make health insurance compulsory but then force insurers to offer an all-embracing, no-restrictions, package at a cheap price. ( If you suspect that there is still some government subsidy hidden somewhere, so do I).
How that would stop people turning up at doctors or hospitals with minor complaints is not clear. It is not clear with any private health insurance, unless you think insurers can exclude treatments. Do you want that?