That isn't the part of the market which is broken though. The insurance companies want their vig, and they have no objection to cost increases because then they get the same percentage of a larger pot, but they're not actually where most of the money goes.

It's really going to drug companies and medical devices companies and doctors and hospitals. Which all still exist even with single payer, and not only that but then have even less price sensitive customers (more unnecessary tests/treatments/procedures) and have a greater opportunity to have their lobbyists turn open the money spigot, because now it's tax dollars -- or government debt -- instead of insurance premiums that customers would balk at and switch to a less expensive plan when the price exceeds a given threshold.

So you still need the very things they're lobbying against. You need something to make patients price sensitive enough to refuse over-treatment. You need some kind of price setting mechanism that works better than giving the most money to the company with the best lobbyists. Single payer doesn't get you that.