Sure, but uninsured people still have access to emergency health care now. If an uninsured person gets sick, they try to ignore it. And then they get sicker and sicker until it becomes a life-threatening illness, in which case they go to the emergency room and then very often the hospital gets stuck with the multi-thousand dollar bill. If that person had access to insurance (from the government, presumably) they could have gone to the doctor when they first got sick, been prescribed a $20 generic medication and gotten better. So, in the long run, government health care would be cheaper overall.
That is the variance is standard in pretty much all ideas. It isn't that one side is heartless or the other side has unlimited compassion/guilt. It is just glass half full / half empty.
That sounds much like the same argument for every brand of socio-economic welfare. "If only we gave them opportunities X,Y, and Z, they wouldn't go around robbing!" I've yet to meet someone in life that truly can't afford health care (maybe buy a smaller house, consider family planning and with only one partner, maybe don't drink so much, don't eat out 6 days a week). Now, of course, we know that say something like public housing doesn't create doctors, lawyers, or engineers. It may create 1 for every 1000 people that pass through them.
The same people that put off medical problems will still be putting off medical problems, and as was pointed out, the ones that were not over indulging in hospital services, now will begin to indulge.
The reason I don't want government controlled health care is simple.
The government doesn't do anything efficiently. I don't say this because I think gov't types are inept, I say it because they have no incentive.
For example, every time a city/county/state runs out of money, do they cut public housing funds? No. Do they cut back on pension funds? What about all the "slush" accounts? Do they fire non-essential people, never!
Do they cut back costs in anyway? Yes! They cut busing, the fire fighters and the police.
You give gov't 10% of your cheese and for this you expect fire, police, teachers, postal workers, public transport, public works (streets, water, sanitation). The gov't comes back and says, "Oh, well we mismanaged that cheese you gave us, so give us more of we'll cut the services you want most". What choice do you have?
My prediction for gov't run health care, given the tendency to cut fire/police/busing. When the system goes broke, and we all know it will, they will cut days on which you can have an MRI, or cut surgeries, and put up their hands and say "support a tax increase then!"