It's much more nuanced than that.
First we must stipulate that the price you pay is not the tuition price. I went to a school that was #1 in tuition when I attended and I paid less than state school -- not because of loans but because of grants the school gave me.
Look at the history: in the 20th century, there was a general understanding that public education was a good thing. My grandmother born in the 20s only graduated middle school. My parents born in the 50s graduated high school and went to a little community college. By the time I was in school in the 90s, it was expected that pretty much everyone who could, should go to college. All of my peers went to college. None of my parents' peers did.
What happened due to this was there were too many students and not enough schools. This caused tuitions to rise. So they built more schools, but that doesn't really solve the problem because as the schools become more popular, the land they are built on and housing around them starts to increase in cost. Schools can't do much about this, but they do have to raise their tuition in response. Schools are in nice areas and attract a lot of high-net-worth professionals. You build a school somewhere and soon you need a hospital, some high tech jobs, a sports team, music venues, public transportation, lawyers, patent attorneys, day cares, and then eventually even more schools. So there's a pretty normal supply/demand issue going on that causes tuition to rise naturally before we even talk about student loans. See: Merced CA over the last 10 years after UC Merced was built there. It went from a Central Valley backwater to a really nice place to live and raise a family. So schools aren't really even schools anymore, they are more like small cities.
What about those sky high sticker prices? Are they that high because schools are confident the government will foot the bill? At the middle tier, yes. These are the schools that are not getting the best students, or the richest students, but they still have to pay competitive salaries for teachers, they still have to pay market rate for land, they have to pay for all the site licenses to Google and Microsoft and Mathworks and Elsevier and Jstor etc... and they still have to deal with the economic reality of being a non profit entity with a public service mission. But not for student loans, those schools would not exist, which would mean higher tuition at the remaining schools.
But, loans to mid tier schools are not the reason Stanford can charge $60k a year. Elite school can charge that, because plenty of rich people are willing to pay full sticker price to send their little bundle of joy to those brand name schools. So that's what elite schools do, they set their price at the level those rich people are willing to pay, and that sticker price keeps going up and up because that elite population keeps getting richer and richer.
The elite institutions set the ceiling, the community colleges set the floor, and then all the in-between state schools, private SLACS, off-ivies, price themselves in the middle. If student loans went away tomorrow, the lower tier schools would suffer most, but the upper tier schools would just admit more rich families and charge them more. Maybe there's not enough to spread between them, maybe they have to close some departments and hire fewer professors, but you wouldn't see tuition fall unless the rich couldn't pay it anymore.