The government controls 75% of the supply (most students go to public colleges). Government employees engage in a massive sustained ad campaign targeted at impressionable teenagers to juice up demand. Then the government issues 92% of all the loans used to pay for the education. The government isn't just "involved" it's deeply embedded on the supply side, demand side, and the middle.
The supply/demand/finance angles are all different governments, though (state/local/federal). It's incorrect to attribute loan issuance and college administration to the same entity, because the federal government and the government of e.g. California are not the same entity.
Also, running an ad campaign is not usually considered "deeply embedded on the ... demand side". Coca-Cola has certainly run massive ad campaigns, but it would be weird to say they're running the demand side of the beverage market.
