I don't think housing costs would drop that much, if at all. What matters is the monthly mortgage payment. Property prices drop when property taxes go up, but mortgage payments tends to stay similar.
In one sense that sounds like free money for the state. However, home equity constitutes the largest share of savings for people, and that's more true the lower down the income ladder you go. So you're taking money out of the working- and middle-class. In some circles home equity is considered a poor savings vehicle, but they don't seem to consider that it's the only secured loan most people can access for leveraged investment. It's the only investment available to most people that let's them leverage capital markets the way the very wealthy can.
And FWIW, residential rents also track monthly mortgage payments, so renters aren't likely to see any difference, either. On average renters pay roughly the same amount per month they'd pay as owners, just without building any equity. I suppose they might be slightly better off after accounting for transfer payments (i.e. public services, entitlements, etc), but they'd be even better off if they could become owners rather than renters.