From many anecdotes I've read about medicine in socialized countries, it's unwise to assume that having insurance is the same as access. Many procedures are made unavailable due to being considered cost effective by The Single Payer, and the wait time for specialists can often be long.
Further, what counts as "costs" in US health industry is kind of funny - health care providers have a book-price that is often inflated, so that insurance companies can claim to their customers (mostly employers) what a good job they do in reducing costs.
For the opposite perspective, some of these points are discussed in much more detail in these two recent EconTalk episodes:
* https://www.econtalk.org/marty-makary-on-the-price-we-pay/
* https://www.econtalk.org/keith-smith-on-free-market-health-c...