If anything it’s existing shielding is the problem. The supply of doctors is artificially controlled and limited which means you can’t lower pressure through capitalistic approaches. Without fixing the supply (more doctors) or demand (don’t treat patients) issue these problems won’t go away.
At least in Germany, there are massive amounts of open job postings for nursing and other care staff. The problem is that the wages are way too low, as the county/city-owned clinics have to adhere to the budget rule of maximum efficiency and minimum taxpayer subsidies (which led to massive outsourcing and other cost saving efforts in Munich [1]) and private-owned clinics are under financial pressure from their owners (you gotta make those 18% EBIT [2] somehow), while at the same time the amount of money that the public health insurances pay for procedures (fixed rates across the country) is not enough to cover these costs.
[1] https://www.kma-online.de/aktuelles/klinik-news/detail/staed...
Medicine is heavily regulated everywhere and people always complain about the same problems in the US and Europe, and these problems are the same kind of problems caused by regulation and guild-like organizations all around the world, in all industries at all points in time.
Even if you were to let someone else pay the bill, it's idiotic to have things organized as they're now.
No, it's no problem there. Raw materials / medicine, consumables, land, energy, work itself - everything has a price and thus a proper socialized healthcare system can account for everything, while putting patient outcome first and not money/profit.
Ever talked with a nurse or a doctor? The reason is simple: way, way not enough staff at hand combined with maximum allowed/planned time for a certain procedure (e.g. 10 minutes for a full assisted shower) which only works in perfect conditions but falls apart under the slightest change (e.g. patient wet themselves), and combined with ruthless MBA-style C-level execs actually firing people for not keeping up with the numbers.
Medicine, especially highly critical / urgent care, needs to be socialized and shielded off from all negative effects of modern-day capitalism if this is to change.