This is just my own personal hair brained idea, but my own idea on the best way to reduce the price of housing is simply to make it harder for people to buy a home. Hear me out. I do not mean increase the price of homes. I mean to drive prices down, you must restrict peoples ability to buy a home.
Essentially, if you make it easier for people to buy a home, then you will end up with more buyers, when you have more buyers then the demand for the product is increased and that results in the price of the product to rise. If you restrict the buyers then there is less demand for the product, and less demand results in the price of the product dropping. If less people are able to buy, then less people are able to sell, if people can't sell then they will lower prices.
It is hard to restrict buyers for houses, the only way I've thought of that is possible is to increase the minimum down payment size. If the minimum down payment size was increased to something really big like 50% for a few years, that would drastically reduce the number of people that would be able to buy a home, and that drastic decrease in buying demand would result in a drastic decrease in prices.
Conversely, where I live the minimum down payment is 5% if tomorrow the gov were to say "we want to make it easier for people to buy a home, and so we are reducing the minimum down payment to 2%" that would mean that more people would qualify to buy a home. More people qualifying would mean that there are more buyers and buying pressure is increased, and increased buying pressure would result in rising prices. So lowering the minimum down payment size, while well meaning, would soon result in the prices for homes going even higher and we would be right back where we started with many people being priced out of the market.
Increasing the minimum down payment size would also mean people effectively get priced out of the market because they would be unable to afford the minimum down payment, but conversely, the prices of houses would fall drastically because of the decreased buying pressure. People would be forced to save for a bigger downpayment, but the total they would pay when they actually buy the home would be decreased.