That sounds fine to me since I come from a family with a good amount of wealth, but I think that kind of screws over all the poorer people a little bit. 

Read what I wrote again.  "...at costs all could afford."   All does include poor people.

Think about this, when you get health insurance from yoru employer, they usually pay a protion of the costs - averaging about $10k per year.  Now imagine that employer simply giving your the $10k to be invested in a medical savings account instead of going toward a corporate insurance plan.   With a free market health care system and a $10k per year medical savings account, even the poor can do pretty damn good with healthcare.  Half of that could go toward standard medical bills (checkups, prescriptions, e tc...) while the other half could be used to buy personal medical insurance with a high up front premium which would cover a ton of things aside from general medical (which you already have covered).

See?  Works out pretty well for everyone.