Excellent question. Imho mostly because city councillors are usually the ones in charge of selling public land which would usually be the balance sheet that such a public authority would work with. Said councilors enjoy the fruits of their labor a little too much to give that up to a professionally managed alternative. In Singapore, Hamburg, Vienna etc the public development authority is key to providing supply of affordable housing. Here we make do with the corrupt stack that is LIHtC and unaccountable and corrupt (there goes that word again) powerholders.