Personally, I look at markets as tools for solving optimization problems, and I'm saying that one of the factors that makes that work well is missing here.

That was never the theory. The theory was that we decided we needed everyone to go to college. We realized that not everyone cold afford to go to college. Giving students loans is cheaper than paying tuition directly, because you'll get most of the money back, so the government flooded the market with no-credit-check student loans. Nobody involved was ever trying to create a functioning market, even an artificial market.

I also don't think it's true that "giving students loans is cheaper than paying tuition directly". Governments are in a unique position to capture broad societal gains through taxation. So I think the cheaper option is just to fund education, especially the sorts we think increase economic productivity, and then tax the resulting income.

That might well be the case. In my experience, people in countries where university is free frequently complain about life-long students ... people who enjoy being a student so much they basically never leave and live off of public subsidies forever, racking up pointless qualifications. I've met such people too. Countries where taking a course requires gaining a lot of debt don't seem to have that problem, or at least not to anywhere near the same extent.

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.

What are the odds the "loan market" will blow up and need a public bail-out, which will be turn out to be vastly more expensive for tax payers than the free fee and maintenance grants provided by other countries?

Virtually zero, because almost all student loan debt is already held by the Department of Education. There is no "loan market" to speak of.

That said I'm not sure how a loan writeoff is more expensive than just paying the fees outright. Presumably public subsidy is always going to be more expensive than a student loan system.