It's an interesting idea and I'm generally in favor of more social housing. I'd be even more in favor of it not being needed at all.

When market rent would be 1,200 yet you pay 800, somebody is paying for that 400. Typically the government, thus tax payers, and it will be subject to political turbulence. Ideally you want so much supply that the market value rent would be 800, as your idea illustrates.

The problem, at least over here, is that the naked price is still very high even without any profit motive. Land is expensive because cities are funded by that. And building that home really does cost 150-200K.

I'm sure you can find some 50K of wiggle room, but at the end of the day it's still going to cost 350K and rent/mortgage will reflect that.