The thing that happened, which you astutely pointed out, is not related to universities at all. What happened has been called
cost disease but is perhaps better known as inflation. The trouble with talking about inflation is that the most commonly-reported figures are based on a consumer basket of goods which have seen skyrocketing leaps in productivity which has kept their prices down, relative to everything else. This productivity increase could largely be attributed to advances in technology due to the tech industry. However, if you look at other parts of the economy which are proven to be resistant to automation, you don't see those same productivity increases.
And that's the problem! Go for a haircut and you can see the effect on a micro scale. Hair salons and barbershops have remained almost entirely unchanged for decades, yet the price of a haircut has skyrocketed. Why is that? Because you wouldn't be able to hire any hairdressers at $2/hour. Haircuts are more expensive because hairdressers need more money to live. Rent has gotten more expensive for hairdressers, just as it has for everyone else.
Increases in productivity due to automation have generated vast amounts of new wealth. It's the overall increase in wealth which has driven up real estate prices and the cost of rent. Replace the hairdresser in the above example with a doctor or a lecturer and the situation remains the same. The services of these jobs/professions have become dramatically more expensive because:
a) They cannot be automated to achieve the productivity gains seen in other sectors.
b) If wages in these sectors were to remain flat then they would see an exodus of workers to other sectors.
c) Some costs in life, such as shelter, are fixed by the availability of land. Land is finite but the demand for land continues to grow. This issue has been discussed extensively here on HN but all of the solutions I've seen are temporary. Ultimately, we could turn the entire planet into Coruscant and we would still see rent prices going up!