I'm not an economist but I'm pretty sure that the price of housing is determined by supply and demand.

Wouldn't housing be more affordable if more houses would be allowed to be built?

Cities seem to enjoy banning people from using their own property as they deem fit. Instead less bureaucracy with faster and cheaper costs for building housing would curb this phenomenon at the roots.

This doesn't solve issues where there isn't enough land (like an island) but in most cases it should work.