The lawyers that are publicly provided are, generally speaking, of the poorest quality, like the shoes at the bowling alley. Having a ****ty lawyer is one thing, but would you really want to go under the knife of a ****ty doctor, or even have a diagnosis from one?
It would be nice to work out a similar system, where low quality doctors are provided "free" to the poor people at the cost of the government (taxpayers). In law, a ****ty lawyer is better than no lawyer. In medicine, however, a ****ty doctor is just as bad as, or worse, than no doctor.
Now you might say, "Well, why can't the government just provide high-priced doctors." If you put those doctors on the government pay roll, the government will try to micromanage how that doctor does his job. Whenever the government is in control of something, it tries to have its cake and eat it too. That is, if a government is in control of providing someone's services, it also passes legislation that forces those services to be a lot cheaper than they would be on the free market. Long story short, if goverment controlled medicine, it would remove the economic incentive for becoming a doctor by forcing doctors to work for peanuts, so that the government could afford to pay them to service everybody in the country.
Despite the claims of some of the goof-balls on SDN who act as if your SOLE purpose of going into medicine is to help people, once the economic incentive is gone, none of you would be willing to pay for 4 years of medical school, let alone devote more than half a decade of your life's efforts to medical school and residency.