It is not so clear, actually, and the example of the K-12 public schools actaully shows it. Nominally, anyone can go to "the government school", but some are better run than others, and bestow more benefits on the students, and you end up moving the bidding war/arms race right over to "whatever mechanism the school uses to decide who goes there".
In the US, that usually means it spills over onto home prices in the best school districts. So, you get the same sorting of students, the same differential education, but people blow their budgets a different way.
This is not much different from the subsidized result.
(You may object: "But the schools can just sort by merit." Well, a) your previous comment said that it's not so easy to find a test that does so, and b) that's just another mode for the arms race: spending money on ways to game/hack the tests: cf, SAT prep courses.)
The same dynamic has happened for all previous kinds of subsidies to higher education: the amount the government pays for (or provides for free) becomes the new baseline, and people have to spend more money to distinguish themselves the unwashed masses who also go that education. Previously, a high school degree was enough, but now college is. With Sanders's plan (regardless of "free school" vs "pay your tuition"), we may end up at "Sorry, everyone needs a Master's now."
Like the site argues, the result of the subsidies is that everyone has to do the same things they did without it, but you also blow a lot of money unnecessarily.
"Here's the government school. Anyone can go there for free."
"Here's $50,000. Spend it at any school you want."
Without advocating one or the other, it's clear that these scenarios are not going to produce the same results at all.
In fact, there's been a great deal of debate about whether or not the second scenario ("school vouchers") is a good idea.