In general, I am wary of government intervention outside the private market (or the capitalist system, as you put it). Local and state governments have not proven themselves particularly skilled at building or operating public housing over the years, so ‘public housing for rentals’ strikes me as a bad idea. Just look at NYC’s public housing system. There are about 178,000 conventional units in the system, of which fewer than 5,000 turned over last year. The waiting list contains over 250,000 families — so a 50-year waiting list. Meanwhile, the system runs a $100m annual operating deficit and has at least a $1b capital deficit, and requires an enormous bureaucracy to keep it chugging. Would the system’s funding and deferred maintenance issues solve themselves if another 175,000 public units were added to the stock? I doubt it.
That said, your proposal for limited equity co-ops is not ‘outside the capitalist system’, but in fact could be completely within it. Providing a tax benefit to a builder could incentivize the construction of these properties. (Or, perhaps, a density bonus if certain units in a for-sale condo building were retained as limited equity.) Using tax policy to incentivize construction is generally a better bet than creating new responsibilities for the government. NYC does this all the time, with programs like the ICAP. So with rental housing, I’m much more inclined to expand the low income housing tax credit program, which incentivizes private developers to construct and operate affordable housing, than I am to try to create a parallel state system.