I don't think that's the theory at all. I think the idea is that if you don't take profits out, there is more money available for healthcare goods and services...
The problem with that "theory" is that it assumes that while the profits are taken out (when you go to a government-run system), everything else stays the same.
Problem is, it may not be that simple...
- Are people who would work for the private insurance industry paid the same as those working for a government-run system? I suspect not... Here in Canada, our public service sector is highly unionized, and as a result salaries are often higher (for things like file clerks, etc.) compared to the private sector
- How much incentive do they have to reduce costs in a public insurance system? We've probably all heard stories about "lazy" public servants, and things like layoffs are rare. On the other hand, I'm sure you're average private insurance company would only be to happy to abuse their employees if it meant reducing costs
So, while going from public to private insurers would save in some areas, it might increase costs in others.
However, almost any system in the industrial world works better than the status quo in the U.S.
Keep in mind that the definition of "works better" is a very subjective one.
It is true that the WHO ranked the U.S. system much lower than the top western countries. However, their rankings were based on several criteria; quality of health care was only one component (costs, equality, etc. were also included.) The U.S. actually ranked #1 in at least part of the criteria for measuring the quality of health care (although they were ranked very low over costs, etc.)
If someone felt that having the option of the "best" health care was the only consideration, they may prefer the American system, even if it meant some people wouldn't be able to afford it.