To follow up on this point, every decade or so via special pleading and political lobbying local authorities get free capital from central government and build houses. They then do very little to manage that housing stock efficiently and just start the special pleading again once the housing list is high and/or the existing stock is run down. There is even a policy to sell it at below-market prices to occupants!
By contrast a fund with an actual cost of capital will work to minimise void periods whether it is dealing with private or HAP/RAS tenants. It doesn't care, it just wants to make money.
This model also de-risks the public sector to a large extent. Social housing demand (location and type of dwelling) isn't going to stay the same forever, and the government purchasing a service on an ongoing basis gives far more flexibility than than having a stock of buildings that aren't the right size or in the right place anymore.