If the average Joe now has more money from the government, wouldn't that drive the property prices up? Polish govt has a program where a mortgage is guaranteed to have 2% interest rate, while in reality the govt pays the difference between the 2% and the actual bank's interest rate, and that just made the prices of housing increase.

The only way not to give money to already rich developers is to have the govt build houses on its own to compete with the developers themselves, which is I assume unthinkable in the US. That would literally be communism