It's the exact same story as with healthcare. If you try to implement welfare by throwing money at the private industry to make their product more affordable, the prices simply grow up (which is basic supply and demand stuff - more people with money = more demand).

For this to actually work, you need to either regulate prices, or have a public-owned, government-run nonprofit with an explicit mandate to keep the prices as low as they can be, regardless of demand - and throw subsidies only at it.

I think this is true with regards to loans and grants that are independent of the university, but not for direct subsidizing of the education through the university. That is, I think you're right that guaranteeing large loans and things like Pell Grants simply incentivizes universities to increase their prices by that much.

However, what should/could be done, is states could do what they've been supposed to do this whole time and begin actually covering students' tuition directly. Unfortunately most states don't increase tertiary education spending YOY at a rate fast enough to cover the increasing costs and enrollment numbers. So the burden falls on the federal government, which gave us this terrible debt-based system