Mike Konczal has a new paper out trying to make a generalized case in favor of "public provisioning" of services and against vouchers and coupons. To take an example he doesn't use, advocates of public provisioning would say we should stop giving low-income families Supplemental Nutrition Assistance Program, or SNAP, cards that they can use at the grocery store to buy food. Instead, the government should operate a parallel network of food depots where people can go to pick up their monthly grocery ration.
As you can perhaps tell from the SNAP example, I'm a bit skeptical of this as a general matter, but I think it's sometimes correct. Providing a single large insurance pool (Medicare) makes more sense than trying to regulate and subsidize a series of competing private entities (Obamacare). But I don't think it's wise to try to make an unequivocal case.
One reason for that is that I think the concepts of "public" needs to be a bit problematized here. In Iceland, the K-12 schools are all run by the government. And they're excellent. And good for Iceland. But if you're a low-income American and you'd like to move to Iceland to enjoy the excellent public schools, you're out of luck. The Icelandic public sector is providing those schools as a public service to the citizens of Iceland, but not just to whomever wants them. That's a whole country, but its population is less than a 10th the size of the population of Brooklyn. And we see that within Brooklyn the same kind of issue arises as seen in the Park Slope public school rezoning controversy. The public schools in Brooklyn are publicly owned, but they don't just accept all comers. Whether you can get into a desirable school has to do with where you live. But the houses that people live in are privately owned.
The recent rezoning "ends several years of hand-wringing over Public Schools 321 and 107, both so desirable in Park Slope circles that home buyers in the area pay a premium to live within the schools’ attendance zones." Which is to say that PS 321 and PS 107 are operated by the public sector on behalf of members of a rather exclusive club of Brooklynites able to afford a particular neighborhood. But you also see this on a broader level. Everyone my age will recall when Andrea Zuckerman lied about her address so as to be able to attend high school in Beverly Hills. That's a whole city that operates essentially like a gated community for rich people.
To make a long story short, while "public" and "private" exist as distinct legal categories in terms of what's morally and conceptually significant, there's a blurry line at different levels of government between a "public" entity and something more like a club. It depends both on the nature of the jurisdiction, its size, and what in particular you're talking about. It's often important to ask who's eligible to use a given public service—or simply who's realistically likely to want to use it—and not just how its provision is formally organized.