Isn’t this already a solved problem with models that are used in various countries in the EU? Where the education is financed through taxes, thus you don’t pay anything up front, but keep paying for it for the rest of your life.
In Australia we have interest free loans, I believe they are issued by the government. You are only required to pay it back if you earn over a certain amount per year, in which case it’s like an extra tax that lasts until the loan is paid back or you earn less than threshold.
It’s not perfect but it sounds like a good starting point.
England has a reasonably fair system where tuition fees are fixed and the loan functions more of like an additional tax only for university graduates (i.e. min income limit, low interest, etc.)
Room & board constitutes 1/2 - 2/3 of the cost of undergraduate school. Even in European countries with free tuition (and that's not as common as you think) students still often must take out loans for living costs. With financial aid the typical American student ends up financially similarly situated to their European counterparts.
The problem of college affordability is arguably another dimension of the housing crisis. You can look at the numbers yourself. Yet oddly I've never seen this pointed out or discussed, not in the media or anywhere else.
In principle an easy way to lower the cost of college would be for public universities to invest in building more subsidized or free dormitories. The problem is that most of the popular coastal universities are in areas where development is absurdly expensive and contentious, even for government.