The interest rates affecting house pricing is because lower interest rates increase demand. Solving the house price issue doesn't mean keeping interest rates high, it can be influenced on the other side, supply. Which is what you've mentioned, the government could build state housing to increase supply of warm dry houses, and this could be a driver of lower prices if done right.

I also feel the need to point out that house prices falling is good for almost all owner occupiers, even though it's unintuative. And since you have to pay off the same million dollar mortgage regardless of if house prices go up or down, we definitely come into the territory of that trolley problem meme with the trolley flattening a bunch of people and then saying it's unfair to those people if we stop the trolley now before it squashes the next group - even though whether it stops or not has no impact on the ones already squashed.