Yes, I think this is the crux of all our education woes currently. I'd be curious to see how the system would evolve if the DOE eliminated the federal student aid program entirely.

Yes, and the reason that there were easy student loans for worthless education is because of laxity in the accreditation system on which reliance is placed to assure that education is not worthless, and because of a historical lack of accountability for frauds on the part of certain (particularly for-profit, who have the strongest incentive for fraud) institutions.

The recent issues at Corinthian and ITT (including the fraud lawsuits against both and the DOE pressure on accreditors, as well as the ultimately DOE action after the fraud lawsuits and accreditation issues at ITT) are a direct result of efforts to end "easy student loans for worthless education."

Who were they accredited by and for how long?

I believe the most common is ACICS. The fact that more attention was paid to it after Corinthian Colleges was how the process that led to this shutdown started in the first place.