The answer to the first question, why should the government run it, is twofold. First, the government won't run it. That is a common fear-mongering talking point. Second, if the government does not step in to change the system, then who will? My oxymoron friend say "Americans will donate to causes and help the poor". Well, some are doing that now, but not enough to help tens of millions of people. And nothing is stopping anyone from doing this - they can do it right now. That is the broken system we currently have.
Illegal immigrant fears are, in my opinion, veiled racism and more fear mongering. First, there is language in the current bills to prevent illegals from gaining access. There is less likelihood that illegals would benefit from the proposed system than they now benefit from the current system.
And for those who think our current system is so great, please realize that lots of americans go to mexico for health care:
http://www.usatoday.com/news/world/2009-08-31-mexico-health-care_N.htm
Also, many americans are outsourcing their health care to india (for major surgeries):
http://www.globalpost.com/dispatch/india/090724/medical-outsourcing-india
Why outsource? Cost. Health care is too expensive in the U.S. And as an added benefit, tens of millions of American citizens are not covered.
Here's a handy "medical tourism" cost comparison chart:
http://www.medicaltourismassociation.com/procedures.html
Note my previous posting that included the National Geographic chart showing cost vs. life expectancy. The following chart compares cost to satisfaction - the U.S. scores poorly on all measured parameters:

"The disparity between how much the United States spends on healthcare and how satisfied people are with their personal health makes the country a unique case. While the United States has the highest per-capita total health expenditures of any country in the OECD, Americans report relatively average levels of perceived health, according to Gallup's Personal Health Index."The above chart and quote come from the following report:
http://www.gallup.com/poll/117205/Americans-Not-Feeling-Health-Benefits-High-Spending.aspx Sphere: Related Content






