The cost problem is from the private providers, not the public insurance.
It's from third-party payment, which applies to both.
Things like the price of pharmaceutical drugs here and their associated patent status are more of an originating source problem than a distribution problem if you ask me though there are some ridiculous distribution giveaways as well (ban on importing drugs, ban on Medicare negotiating Rx prices, etc). Leaving the methods of healthcare delivery with private providers, even with public funding, gives those providers the real leverage when the government isn't willing to compete directly. Allowing for-profit hospitals to receive public funding is just as bad as allowing student aid like Pell grants to be made accessible to for-profit colleges. They're going to look at Medicare/aid as a
business opportunity.
I think public funding should be restricted to public facilities for things like this so there is no underlying price increase incentive for private owners. Like the difference between public housing subsidies for privately owned residences (Section 8 vouchers) vs a
Red Vienna-style public housing system or private for-profit hospitals&payments+public insurance for the poor/old vs the NHS. A lot of what is being subsidized in the private provider/public finance system is profit. The price of these services could be lower if we decided that not everything has to be for-profit, that housing shouldn't have to include thousands of dollars a year in profit through the rent because the land is privately owned and the owner wants a return on their investment at least equivalent to the other market opportunities his/her land could make them, that medical care shouldn't have to be delivered as a commodity at publicly-traded market rates but instead at-cost, that money shouldn't be made like this when it comes at the expense of people's standard of living, etc.
I don't think the progressive position should be to say let's ignore the past price increases and focus on limiting the rate of
future increases to at or slightly below inflation or increased tax revenues/personal income. The progressive position should be to cut down those past increases to get to the lowest possible price for those essentials so people can have a higher standard of living and more economic security. Given the way things have gone in the recent decades with the upper consolidation of influence among the rich, those price decreases would probably lead to massive layoffs and drops in profit in the real estate and healthcare industries so they obviously lobby and buy-out government(s) with their power and wealth to stop new directly-government-run services and work to takeover the existing ones (see the steady privatization of the NHS in the UK).
Private service costs don't seem to be scrutinized the same as public service costs, even when they are the only provider. I hear a lot about reducing public costs, but not much about reducing private costs. When is the last time you've heard of a rent cut on the scale of these huge tax cuts? Price decreases don't benefit everybody the same, especially not the people who make money at our expense and I see restrictions on the government's ability to provide services like health insurance/healthcare or housing as signals for upcoming private cost increases in those services, embedded and normalized as annual 'cost of living' increases. Those transfers of income and wealth upwards is what I believe is the mission of neoliberal deregulation and privatization of public services.