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This isn’t true if the government generates more via subsidies.

That will just push down the market rate so that it's not below market rate any more, won't it?

I don’t think so, why would that happen?

I would expect it to actually increase the market rate because setting aside some amount of the supply at below market rate / subsidized means that there is a smaller remaining supply than if they hadn’t done that.

Market rate is the price where supply and demand are equal. If you set price below market rate, the demand will be higher than the supply. So if you create enough below market rate housing that there is no longer a shortage, then it must mean that the supply and demand are equal at that price, which means that it is no longer below market rate.

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