Colleges just raise prices because they know the Government will pay it. If we had college loans setup like other countries do single payer, where the Government refused to grant loans to students going to colleges that raise their rates too much, those colleges would immediately cap their tuition rate increases.
If the Government got tough with colleges, you'd see instant turnaround. Start with the worst offenders and start working your way down colleges. There's no college in the US that wants to lose federal loan money, so they'd all immediately cease these massive rate increases.