yes, your plan would require extreme levels of oppression. it's a non-starter in anything remotely resembling a democracy. and if implemented, the outcome would still not be what you expect.

houses get more expensive due to changes in scarcity, i.e. the number of people who want to buy houses (and can afford to) in an area increasing faster than the number of people who want to sell houses there.

the main lever a government has, is to make new construction easier and cheaper -- that increases the supply. commies don't seem to like that, often because they're NIMBYs, but also because new construction tends to be in the high end segment. commies don't seem to understand that people who move into newly built high end housing do so by moving out of existing, on average slightly less high end housing, which then becomes free, and so on... all the way down to the lowest end housing. increasing supply at the highest end increases housing supply in all segments, though with increasing delay.

an option that doesn't help much with existing scarcity, but with future changes in scarcity, would be a recurring land tax (proportional to the local value of owned land). in order to be democratically feasible this would have to be introduced very gradually (over the course of 30+ years) and combined with tax decreases elsewhere.