Would it require, no.

Would it be a good political move to do so, probably. Though attempting to cut down the US for-profit health insurance industry with its massive lobbying arm is already unhealthy for a political career so maybe the ire of thousands of unemployed middle managers and cubicle workers wont matter.

> Universal healthcare is publicly provided ... healthcare

Not necessarily.

Australia's healthcare system incorporates a lot of private healthcare providers, on the lower level (GP's, dentists, etc) to my knowledge it is mostly private.

Private providers negotiate with the public insurer, Medicare, for how much they are paid per medical procedure and any extra they charge to the patient or any supplemental private insurance they might have. Some aim to charge entirely within what Medicare provides (called Bulk Billing services) while others charge more.

Because Induced Demand exists and so it will always be impossible to build enough transit infrastructure to solve congestion... and private cars are the most expensive, least scalable, most damaging, least equitable and socially isolating mode of transit to build more infrastructure for.