K12 education, Medicaid, Medicare.
(cue loud tedious misinformed discussion of how terrible these programs actually are, but compare with similar private sector offerings and the truth is painfully clear)
Go to any major city in the country and you will find that the K12 education you cite is not doing so well. (Great example of failure is Baltimore, which spends more per capita then any other city and it has the worse public education system in the country.)
Additionally, you will find in the majority those cities, most are for school choice and charter schools precisely for these reasons, and that the charter schools are performing better.
Yes, there are some bad charter schools, but many are doing better.
Now the reason for this is not necessarily a different product or more money or being more efficient, but because if you care enough to send your child to a charter school, you are obviously more involved in your child's education then those who do not put forth the effort. This, parental involvement, accounts for over 90% of why a child does well. This has been studied throughly and proven several times all throughout the world.
Now, would this work with public schools, absolutely, but no politician talks about this. They only concentrate on getting more funding, which completely ignores the actual solution. (It is not a surprise of course, because how would one get reelected if he spent his time telling parents how bad of a parent they are being. It would be political suicide; just look at what happened to Obama when he pointed out black fathers needed to start being real fathers. Regardless though, this leads to an ignorance on the actual cause of the problem that ultimately prevents the solution from even being discussed let alone implemented.)
This is why the major cities never actually fix their education system.
Now you could ask why do we not have charter schools in rural and suburban areas and why do the K12 programs work better there? Because, if you live a rural area, there typically is not a market for more then one school, so the more involved parents send there child to the only school available, and this helps skew ups the results. For suburbs, the cost of living tends to be more, lending to higher incomes, and higher income people tend to be more involved in their child's eduction. Also, many suburbs are also spread out over a larger geography that makes the market for charter schools slim. This essentially creates just one choice as well. In the suburbs where it can work, the incomes tend to be high enough that the cost of a private education is not prohibitive for enough to make the private school model work, thus canceling out the market for charter schools even more.
I will let Russ answer the Medicaid and Medicare part since he knows more about medicine then I do.
PS. You could look at this and point out that since the difference is parental involvement, and if you are an involved parent, sending your child to the public option would result in the same outcome. The actuality of this proposal is debatable; I would argue that a lower performing school tends to be more violent and distracting, which takes away from the level of eduction. However, how people perceive the quality of the education gained from a better performing school is going to be higher then that same exact education from a lesser performing school. This would effect the future options one would be given from those who look at these things, such as employers and college admissions boards.