"...... the whole thing collapses"
So, gov't subsidies are what sustain the housing market? That, in itself, is THE problem. Gov't subsidies distort the market and mislead people. The subsidies exaggerate natural market swings. A stable housing market will return when the gov't stops meddling.
Housing prices rise/fall even with the subsidies in place. The subsidies do not prevent home values to drop. The subsidies only increase prices above their free market level. Housing prices WILL decline when the subsidies are removed. That is why there is so much opposition to eliminating the mortgage deduction; current homeowners will see their equity decline.