One pedantic quibble. It's not necessary to build cheap housing specifically. It's perfectly fine simply to build more housing of any type. Today's new expensive housing will indirectly create more cheap housing in the pre-existing housing stock, by pulling demand out of the market.
Throughout all of American history, new housing has almost exclusively been relatively expensive housing. The rich prefer to live in new housing stock, and the poor prefer to get discounts on old housing stock. The same way that a robust new car market creates a secondary market for cheap used cars.
A lot of people with good intentions will try to block any type of development that doesn't specifically include affordable housing. But this is the exact opposite of what's needed. The best way to lower the cost of housing is to increase the supply of housing. And the best way to do that is to unleash developers whatever is most profitable whenever and whoever they want without government regulation and NIMBY activists getting in their way.