You get an insurance card that you can take to any doctor that accepts Medicaid, they'll treat you for free or for a nominal (say, $5) copay and bill the state. Mental health treatment is covered, prescription drugs are covered, the only major thing it's missing, as far as I know, is dental. But the main catch is that many doctors don't accept it, since Medicare generally reimburses at a lower rate than private insurance. Anecdotally, I was on Medicare in the rural Midwest several years ago and I think I had two choices of GP within a 25 mile radius.
Everyone doesn't have it because it's means-tested - if you make more than very roughly $1,200 a month you don't qualify. You still qualify for income-based subsidies at that point (under a totally different government program), but at higher income levels the expectation is that either you pay for your own health insurance premiums out of pocket, or your employer pays them for you. It's all very complicated, but that complexity is the price we pay so that we higher-income Americans can say that our employer is paying a "premium" and not a "tax." Evidently some of us care an awful lot about that sort of thing.
Everyone doesn't have it because it's means-tested - if you make more than very roughly $1,200 a month you don't qualify. You still qualify for income-based subsidies at that point (under a totally different government program), but at higher income levels the expectation is that either you pay for your own health insurance premiums out of pocket, or your employer pays them for you. It's all very complicated, but that complexity is the price we pay so that we higher-income Americans can say that our employer is paying a "premium" and not a "tax." Evidently some of us care an awful lot about that sort of thing.