Single payer doesn't really matter. There are countries which do great without it (Switzerland for example). If you want cheaper stuff you need to focus on getting rid of cartels and anti competitive practices. Train more doctors, some through shorter track so they can provide simple checkups/treatments. Force everyone to list prices in clear way. Change the law so you don't go bankrupt when something goes wrong and that you don't need several million investment to provide something as easy as MRI scans. There are tons of obvious things to do. Single payer will not solve those problems.
Full blood panel with all hormones, cholesterol, mineral levels costs like 80$ here and I can get it done any day, any time by just walking into a lab and telling them what I want to get done. No doctor consultation, nothing. The results are available online the same day (with the exception of things like free testosterone that takes several days as they ship it to outside lab). This is done without any coverage, insurance or anything. Just paying out of pocket. Think about why it's possible in a poor Central European country and not in US. Hint: it's not what progressive politicians tell you.
The US has a market for private healthcare, but healthcare will never be a free market. It's the antithesis of a free market. Intrinsic inequality of bargaining power and information asymmetry are the biggest reasons for this.
This point is usually brought up in regards to healthcare emergencies. But emergencies only make up a small fraction of healthcare spending[1].
Maybe people have less bargaining power because of agreements with providers/insurers. That's a barrier to an effective market with price sensitivity, but it's a contingent one, not a mathematically inevitable one. For much of our healthcare spending the balance of bargaining power could look like it does for food.
1: https://www.politifact.com/factchecks/2013/oct/28/nick-gille...
> How do you explain nations with more government involvement in healthcare who have lower healthcare costs?
They made other tradeoffs and inverted the graph. For example cellphones are way more expensive on some countries, cars are taxed up to 100% on others.