My house was build some 30 years ago by students who paid for their education. They added a new badge of construction workers every year (and the oldest badge got their diploma). I have no idea what happened with the concept. They are now back to [the usual] building things next to and inside the school then demolishing it.
I imagine the collective mind could greatly benefit if everyone had to spend a few (education focused) months in construction (and/or say farming etc) Its not boring if its just a short period. In construction you can learn the basics of a discipline (and make yourself useful) in just a few days.
But I agree it is the obstruction that prevents cheap housing. Its probably intentional (to some extend) simply to drive up prices.
How could social housing fix the issue? The underlying issue is too many obstructions to building new houses, which restrict supply. If commercially motivated developers cannot build enough houses due to bureaucracy and obstructions by NIMBY groups, then municipality would likely fail too.