Funnily, it's because of the massive amount of bureaucracy involved.
I say funnily, because the US government is know for being light on bureaucracy. Yet, somehow the US healthcare system has given it a run for it's money.
Every hospital has a significant amount of staff dedicated to billing and negotiating with insurance companies. Every insurance company has a fleet of employees who work on optimizing costs, denying coverage when possible, and negotiating bills with hospitals. And on top of that, a large part of the job of HR for employers is managing healthcare benefits and negotiating with healthcare providers.
Layers upon layers of ultimately waste.
A single payer system ends up saving cost because.
A. There's no negotiating bills with the government. You either accept what they are paying or you don't get paid.
B. Because the government deals with every hospital, they've got very good insight into what things ACTUALLY cost. That makes it hard for a medical provider to play fast and loose with the numbers.
In otherwords, admin costs for Hospitals go completely away and the government admin costs end up being a fraction of what the rest of the insurance industry is doing (because a lot of what private insurance is doing is trying to make money off of premiums in the stock market). Government does not have that incentive in the slightest.