On the contrary, increasing mortgage mobility should lower prices.
In most cases the mortgager is selling their old house and buying another, keeping the same supply/demand balance. But in some cases the mortgager is selling one house and building a new one, increasing supply and lowering prices.
In theory, but due to various issues such as labor costs, zoning laws, permitting costs, and agglomeration of economic opportunities, that may not be happening at a sufficient rate in a sufficient number of locales.
The Feds do not have much control over those parameters though, so they play with what they (politically) can.
In most cases the mortgager is selling their old house and buying another, keeping the same supply/demand balance. But in some cases the mortgager is selling one house and building a new one, increasing supply and lowering prices.