While this seems obvious on the surface (if there are more houses than people in some area, prices should drop right), are there real world example of this working that way?
More housing just creates more demand which drives more density and yet more demand. The densest area in the US is Manhattan, not known for being a cheap place to live. Second densest is SF, also not a bargain for housing.
I doubt there is a simple one-way causation, it's a feedback loop.
The economy of a city only grows if more and more people and businesses move in, which means more housing and density which in turn grows the economy and prices even more.
Is there any example of a city where housing was quite expensive and over time housing became cheap merely by building more of it, while still remaining a thriving city?
As far as I can see, once a city is thriving and expensive it only becomes cheap through economic collapse (e.g. Detroit) which isn't the solution anyone wanted.
Either induced demand is real which implies if you bought a corn field in Idaho and built enough high density housing you could make the housing more expensive than Manhattan and walk away richer than god.
Or high demand drives housing density and high prices.
For an example how trying to build something up artificially in the middle of nowhere can fail, see California City. OTOH, for a counterexample see Las Vegas.
> Or high demand drives housing density and high prices.
Yes, and it also drives even more demand and even higher prices.
The premise of the OP is that you can make housing become cheap in an expensive city by building a lot more of it. If that works, there ought to be a few success story cities where it happened. Which ones are they?