There's no direct comparison to be made with housing shortages. The only scarce educational resources are those that convey high prestige, and there's no way to manufacture more of that through higher tuition payments because it's the scarcity that makes it prestigious. Harvard would be just as prestigious if it were free to attend as if it were $100k/yr to attend. Whereas in housing, if you had people lining up to pay $100k/yr, you'd be throwing up new buildings as quickly as possible.

In general there is a an oversupply of higher education capacity due to demographic downturn and a decline in enrollments since 2010. Smaller colleges have been closing because they cannot attract enough students.