Then, set up clinics where one option for care is First Aid. People who don't have insurance and want to restrict what they pay would have the minimally priced first aid, self-care option to choose.
When all that happens, insurance companies can do what they do best. That would be to scare people into being insured, and to make prices and options attractive so that people will sign up on their own.
Those who can't afford much? Perhaps give them a stipend with which they can make the same choices as everyone else, only have it be enough to only cover catastrophic illness plus the Fist Aid option. I would wager a whole lot of them would pocket the money and sign up for the First Aid option. Anyone who does nothing will have that automatic policy.