Housing authorities just need to lower housing prices. The model already exists there's just a lot of money fighting it:

1. Build large 5-6 story buildings with dozens or hundreds of units in areas where prices are too high. These buildings should be mixed use on the first floor with no car parking.

2. Sell or rent them at well below market rates to people that currently do not own homes anywhere under the condition they only sell under those conditions as well

3. Watch the market prices fall.

There's currently enough housing in most places there just needs to be a needle applying downward pressure.