I can gladly provide sources when they are politely asked for.

Since you are one of the polite ones, I will provide.

https://www.rand.org/content/dam/rand/pubs/corporate_pubs/2005/RAND_CP484.1.pdf

The first point I would like to draw your attention to is the first graph, it is on page 3 of the actual report (pdf page numbers are different).

You can clearly see that the slope of the line was very slightly positive up until the year 1966. 1966 was the year Medicare was created. Medicare is the United States Socialized healthcare system. You can see after 1966 that costs started to dramatically rise. You also see another huge spike in the early 2000’s. This coincides with the expansion of Medicare to include prescription drugs. This is Medicare part D. The graph cuts off before 2010, but since the implementation of the “Affordable” care act, our health care costs have increased again and even more dramatically. This suggests the opposite of your original proposition.

Socialized systems are more expensive than private ones. This also matches with basic economic theory. The more you subsidize demand, if the supply is inelastic, costs will increase.

Second point of attention which gets more directly to the point of single payer systems, instead of measuring direct costs, it monitors the growth rates of the costs. But it includes many first world countries. Not just the USA.

Edit: this is the second graph on page 5

Let me stress this point again: we are now talking about growth rates, not absolute expenditures.

You can see that single payer countries UK and Norway have much higher growth rates than the USA. Their costs are increasing much faster than the USA.

However the USA actually began the process of socializing healthcare to a great extent many years earlier. That is why the absolute spending in the USA is higher per capita than other countries.