I've wondered lately, since the government backstops both banks and mortgages, shouldn't it just cut out the middleman? It could either give cheaper mortgages or use the difference as tax revenue.
There's this weird brain worm from the 1980s in a lot of western democracies where everything government does has to be bid out to companies and driven by profit motive under the theory that it produces better outcomes. It doesn't seem to be working after decades of this experiment, but people keep trying.