The government is not a business.
That's kind of a non-sequitor; downstream income has nothing to do with the cost of the good. Somebody could quadruple the price of steel, but people down the line will still make income.
Student loans is a great example here. Government comes in, gives everyone financing, and the price of college just so happens to rise to the level of financing the government will offer.
I don't see a way this ends well. If they offer it to everyone, housing prices simply factor that in. If they only offer it to the very poor, it just collapses two existing housing brackets into one (the very poor can spend like they're only kind of poor, making very poor cease to exist as a category and lumping the existing "very poor" housing into the "kind of poor" category).
We need more housing or lower demand. That's the solution. Anything else is just shuffling around hardships.