I would actually say it can only scale linearly or slower than that. Either you gain efficiency due to economies of scale, or worst case, you make 10 smaller systems.
But I still don’t see how the scaling would be worse than linear.
How is managing 300 million people more than 3 times harder than 100 million people? The effort per person shouldn’t increase, and everything common scale lower than linear.
10 smaller systems then require a layer on top to manage those 10 systems. Now you have 11 systems. And there definitely won't be only 2 layers in your public healthcare system for 300M people across 50 different states.
Have you never worked in a large corporation before and seen all the intermediary layers of beauracy? If you have, imagine the largest company you've ever worked for, multiply it by 1000x, and then imagine but there is no profit motive driving efficiency (or competition) and your customers all demand access to a resource that is finite. Except they're not really "customers", so you can't say "sorry no more product left, better luck next time".
That is publicly funded healthcare at scale.
What "economies of scale" do you imagine centralized publicly-funded healthcare has?
But I still don’t see how the scaling would be worse than linear.
How is managing 300 million people more than 3 times harder than 100 million people? The effort per person shouldn’t increase, and everything common scale lower than linear.