Without student loans, only children of the wealthy will be able to go to college.
By that logic, the day that student loans end, all of the colleges will go bankrupt. No, obviously they will lower their prices until low enough that enough people can afford it that the college is solvent.
But the Federally subsidized student loan is a subsidy: the interest rate is artificially low, it can go unpaid for decades, the lender charges the government fees based on the amount of the loan, and the government guarantees at least part of the principal. Since the university and lender have incentive to increase the tuition cost, and the students have little incentive to find a lower price, the inflation occurs. It seemed to start in the 80s, but it would be difficult to determine the exact date it began based on this graph.
Also note that ending federally subsidized loans does not mean that there will be no student loans available. They may be more difficult to get, and they will have higher interest rates. But these will keep the tuition cost nailed roughly to the rest of the market.