If anything, the billing abuse is a much, much larger problem than lack of insurance. At least, that has generally been true since the ACA/"Obamacare" changes passed.
My thought experiment for some of this. Suppose the US government had a rider on every insurance policy in the US that said the government would pick up the tab for any amount over say $100,000 (think catastrophe insurance). Now, you would make the assumption that looking at the actuarial tables this would reduce the cost of health insurance because no insurance company is on the hook over $100,000 (thus no need to worry about the million $ payout).
I am more and more convinced that hospitals would screw this up with over billing for every damn thing and every visit requiring a stay would get jacked to > $100,000.
We have a solution. It is called "make it a personal criminal liability". Go after individuals. No matter how low they are on a totem pole and no matter how high they are on a totem pole. Committing such fraud should lead to bankruptcy ( all assets wiped out ) and jail time.
You would be amazed how quick those "accidents" and "mistakes" stop happening.
Unfortunately, as the society we do not want to throw Suzi the billing clerk into the slammer, which means that Mike, the Billing Manager, does not get a slap on the wrist, which in turn means that Jack, the VP of Billing Revenue Optimization, does not get Jackie his wife go bananas on him when their bank accounts, house and kids college fund is seized which means that Jack is represented by the public defender and ends up in a slammer together with Suzi, who actually pushed the buttons, Mike, who told Suzi to do it, and Jack, who came up with this wonderful idea.
Socializing this theft turns it into Just Another Crisis among the hundreds of others the US has.
The "true" purpose of the ACA was to do something about spiraling health care costs. What was passed was a "compromise" where the people pushing for a public option got nothing and the most onerous restrictions on the monied interests were rolled back. This has not been an effective solution in many senses. I decline to further characterize the issue, however; this is extremely close to a political discussion, and those are ban-worthy here.
I would not put too much weight on conspiracy theories though. The healthcare system is massively complex. It will take multiple reforms to eat this elephant.