That is such a naive post, I don't know where to begin. I guess I'll bypass the "government doesn't do anything well" BS (firefighters, US Mail, EMS services, public water/sewer systems, uncountable other examples prove you wrong)... But, apparently, every industrialized country other than the USA either has free healthcare or a hybrid system like Australia with a combination of free care plus private.
Let's move on to this mythical health "market" you mention. Markets require competition to work. Most areas only have one or two hospitals within the geographic region which can provide most health care services. That's not a free market... it's a monopoly or oligopoly. Monopolies and Oligopolies require government oversight because they tend to abuse their power. Granted, the Dentistry market is far more competitive than say, thoracic surgery.
Still, Insurance isn't a market either. It's also an oligopoly situation where you have to have one of the major carriers to have health providers ACCEPT the coverage you have -- and picking an insurance carrier may give you perks with one hospital or other health care provider, but none with another, so this also limits your market choices. Health care providers are not required to accept your insurance.
So, let's talk about pricing - you won't find it listed most places. It's complex... it's deceitful - intentionally. If you have no insurance, you have one price. If you have insurance, it's another price. Then, when billed, you pay a different amount and the insurance company pays the rest -- but not actually. You see, the insurance company negotiates the prices. Say you have a bill for $100K. You pay $5K, the insurance company pays $45K, and the rest just goes unpaid, yet considered to be paid in full. Another individual who has no insurance gets the bill for the full amount - OR if the physician knows in advance you have no insurance will sometimes negotiate a different price - sometimes much lower than what they'd have gotten from the insurance company.
Doctors HATE the insurance companies. They have to hire lots of staff for medical coding to report correctly to insurance companies, fight with them over the billing, and often get paid late -if at all. Doctors also have high malpractice insurance bills and high medical school loan bills. Many other countries don't have these issues -- they even send their doctors to medical school for next to nothing - imagine that! It drives the cost of being a doctor down, increases supply of doctors and drives the costs of medical care down along with it.
The USA medical system is a mess. I'm not a doctor myself, but I have many family and friends in the medical field. They would LOVE a single payer system to simplify everything. They could have less staff because there's no need to deal with multiple insurance companies, less confusion on pricing, and more customers as everyone is covered. Government health insurance doesn't have to be government run healthcare - just insurance. Why have thousands of companies complicating everything when one agency could give you insurance right out of your paycheck with your taxes (just like a company benefit would), and you're insured everywhere for everything except cosmetic surgeries beyond dental. But, I digress.
I'm not sure what dental plans you're concerned about. Most don't cover things like crowns and Hollywood veneers. Most cover regular checkups and fillings - maybe braces for kids if you pay extra. That's not a huge burden on the USA economy... not with 15 Trillion in debt - mostly spent on the military.