Subsidies or government backed contributions won't be the answer. Universities have a societal function and could conceivable operate without profit in mind, but they still chose to be greedy and act like for-profit businesses once government backed loans became a thing. The government tried to fix access to education by directly injecting cash via students instead of fixing it through the universities. Taking the same approach with home buyers would be a mistake because real estate obviously is meant to be for-profit and acts that way, so they will obviously raise prices as well.
So you'll probably need to fuck over a lot of banks, investors, construction and real estate companies to get housing prices accessible to the wider population and change economic interactions in that market. I wouldn't count on that though - the government will probably choose to fix it through backed loans or subsidies again, if they do anything.