Just because something is nonprofit doesn't mean nobody's making money. Private and "for profit" are not synonymous.
Think of it this way. Where do people get better care, have better technology, access to alternative or cutting edge treatments, etc.?
One thing I didn't specify in my original comment. When I say private, I more just mean non-federal government run. I would consider the numerous university hospitals throughout the country private sector, although they are technically government owned if they are state universities.
Having grown up around Iowa City and hearing from various sources, I'm 100% certain the UIHC is better resourced than the VA hospital down the street.