Because it is (close to) impossible to write an objective measurement system that can tell the difference between a legitimately good school (as defined by the rather abstract things that we as a society want education to be) versus a school that is good at gaming metrics. And when you introduce a direct profit motive tied to metrics, it is highly likely that executives will find a way to maximize profit while minimizing cost, which likely means gaming the metrics and doing little else.
Government institutions and private nonprofit institutions have this problem already (at least somewhat) with U.S. News and World, but they are at least theoretically controlled by foundations/trustees/the state, which have in common the property that they profess an interest in education itself, not education as a means of extracting money from people.
I don't really see anything you've pointed out that doesn't apply equally to profit / non-profit. Given your second paragraph, I'm not sure you have a lot of hope that public schools are doing the right thing.
"they profess an interest in education itself, not education as a means of extracting money from people."
I'm sure counter-examples exist, but the best businesses are not about "means of extracting money from people", that is just a natural consequence of a decently provided service. Governments and Institutions also think in terms of "extracting money from people" although they are less honest about it (why yes, I've worked with both).
Well-regarded nonprofit colleges are mostly doing the right thing, with some stat-gaming around things like number of student organizations, expenditure per student, etc. to jockey for higher positions in U.S. News. Not in their core educational functions.
We as hackers can tell that our $1,500 Macbook Pros are better than $450 Inspirons, so Apple thrives.
The problem is that it is much more difficult to distinguish between high-quality and low-quality education. With vocational training it is easy: do employers like the graduates? Do they get hired at a substantially higher rate than non-graduates? Do they get paid more? But with education it is much, much harder.
There are some objectively observable traits that can (sort of) tell you whether an environment is likely to contribute to certain kinds of personal intellectual growth: do students read and respond to (at least) the Western literary canon? Are classes small enough to facilitate meaningful discussion? Are a wide variety of courses taught? How deep into their respective fields to the syllabi go? Does the school actually own the equipment and materials necessary to perform the kinds of laboratory procedures that a graduate in the sciences should have experienced? Are the classes hard enough that not everyone gets perfect scores all the time? How many hours do students need to spend on the coursework to make good grades?
But all of these can be gamed. Hire lots of cheap, incompetent professors. Have small group discussions where you basically just sit there for an hour talking about nothing and wasting time because the professor is doing something else on his laptop. Force students to write papers, but have an institutional culture where they are mostly bullshitted or plagarized and no one cares. Don't actually take the time to grade students' work, but assign bad grades to the handful of students least likely to make a fuss. Assign lots of busywork. Etc. Attract students with good high school test scores by lobbying to divert government funding that would have allowed them to attend better institutions, and set your own prices cheaper.
Think of any other property of legitimately good education, someone can probably find a way to appear to have that property more efficiently than actually having it.
The only reason this isn't widespread is because administrators are generally people who want to provide good education and have the freedom to spend pretty much whatever they can get their hands on to do so.
If you replace them with administrators who want to serve shareholders instead of students, they will find ways to hit the metrics that actually influence reveneue (i.e. rankings) while spending less than the other guys, and that will probably mean compromising the quality of the education they provide.
I guess I wasn't really clear in my post, sorry. My basic belief is that government / public institutions aren't anymore "holy" then for-profits. The profit motive is transformed into a power motive in public institutions. We can see this with the growing bureaucracy at our public institutions that has raised tuitions and removed resources from students. The quest for grants is in full force and professors are teaching less. The things you list that for-profits can do are done at public universities.
"If you replace them with administrators who want to serve shareholders instead of students"
To the point, I don't believe a lot of administrators are thinking of the student first. I see a lot of "integrity of the institution" which leaks over into free speech and open discussion suppression. Admins do not serve the student first, they serve the politicians and the next step in their career.
As always, their are exceptions, but I don't see many of the dreamers and idealists left.
Government institutions and private nonprofit institutions have this problem already (at least somewhat) with U.S. News and World, but they are at least theoretically controlled by foundations/trustees/the state, which have in common the property that they profess an interest in education itself, not education as a means of extracting money from people.