I agree that there is no guarantee that building housing lowers prices, other causes (like increases in demand) may cause prices to go up (and the increased supply would lower the upward pressure). But I have a hard time to understand how increased supply would not have a downward effect on prices.
It does, if that is the only thing that happens. If land owners or existing home owners willingness to sell, prospective renters or buyers willingness to buy, or any number of different things also change then those effects can overshadow an increase in supply. As they frequently do.
Starbucks keeps opening new locations but that doesn't say much to the effect that the price of a cup of coffee will decrease.
We can build housing much more efficiently and cheaply than we could 60 years ago, yet it's never been harder to buy a house.
The law of supply and demand only works as expected when used in an extremely simplified model.
If you account for all the other real factors (corporate greed, governmental intervention, NIMBYsm) that are skimmed over by economists, it becomes clear that the law of supply and demand should never be used alone to forecast future economic conditions.
If that was true, buying a house today would be as easy as it was in the 60s. In fact it isn't.