Just need price controls (at least limited ones, and the credible threat of more) plus a good public fall-back option, and strong standards for what health insurance has to cover. That seems to be the minimum (Singapore model). Those elements seem to be the common factors in basically all OECD healthcare systems outside the US. Taking it farther can work, and may even help, but those seem to be the things you have to have (though price controls can also be implicit, via monopsony). “Credible threat of more” for the price controls may not work in places less authoritarian than Singapore, so you may have to swing closer to the price-book model of places like Japan, if not just go single-payer.