Eh, a significant portion of the first problem is solved by simply having single payer healthcare like most developed countries (except the States) has.
This is a big part of the problem, but I don't think it would fix everything by a long shot.
In the Netherlands and Switzerland, all health insurance is provided by regulated private payers.
You're gonna have to work a lot harder than you are here to make this point pass muster. It's pretty blatantly untrue.
Yes, supplemental insurance is often a thing that exists. It's generally nothing like the US system's insurance though.