Actually it doesn't legalize cheaper housing, as it won't change the supply/demand. The house prices will increase by the subsidy amount. All this subsidy does is that it funnels more public (tax-payer) money to house owners.
I agree that monetary subsidies tend to simply get capitalized into higher prices, but if you read the article the proposal is to actually build 2,000 units.
What? No. They are building housing and will rent it out below market price. No one is given a monthly check to spend on housing or anything like that.
These programs typically have a formula to set prices based on income. It's a lottery, not a direct subsidy like school loans.