Specifically, I am talking about what is a bachelor's degree in the US--a 4-year-degree that offers you few special, unique, or licensable skills, unlike a masters' or doctorate or trade schools.
In my view, this field of higher education is mostly about gating access to a small job market--the supply of jobs is so limitted that people are (decreasingly) willing to go into debt for a crab-in-bucket's chance at some employment. It also perpetuates class divisions, more now than it did when getting a degree was a real ticket paid non-physical labor.
It's also a highly extractive industry in its own right--price of higher ed has outstripped inflation for over 30 years while professors are being paid less. People are entering there working lives deeply in debt. This is a systemic issue.
The answer, I think, is not simply gov't paying for tuition because that does not deal with runaway costs.
Higher education has a priveleged place on the left, since its where many of us gained a broader worldview. But setting that aside, the institution itself qua institution is deeply problematic in the current system.
My only unique contribution to the topic--which I have not heard anywhere else--is removing higher education as bona fide occupational qualification without justification as to specific content or skills w/o other options to demonstrate mastery. That would leverage current law (in the US) to dethrone higher-ed supremacy, because we know most jobs that require or prefer a bachelor's degree don't practically really require an education that takes 4 years.
Another idea would be to include student debt in bankruptcy. The problem is that we are generating a generation of slaves who will never pay their college debt, so it is the student that bears all the cost for the lies and cost overruns of the institution.
If--and I think Obama started to do this--colleges would not get their money, or banks would not get their money, there would be insituted more rigor into the loan application process built entirely around the probablility that the student would be able to make enough money to pay it back. How many of those loans would be issued today?
We'll see how the enrollment drop shakes out; I think you're right, the main effect will be a more stratified society as only people with cash will be able to get a higher education that will give them access to higher paying jobs.
I don't think things will go as smoothly as you think. Society is toxically positive and children are very suggestible. We give children who are not old enough to drink the power to take on life-changing debt. "What do I want to do for work?" is a hard question to answer when you have no demonstrable skills and no job experience, making college an attractive option for people without family connections in the trades.
I'd also add that the federal government DOES subsidize education to a high degree, and perhaps as much as it does so is how much it contributes to the theft of alumni.