I think there are many examples for induced demand for apartments: People definitely move to get a few minutes closer to work, get a bigger apartment, or just to be closer to friends and family. That means, they will indeed come if you build it.

In my view, housing problems are a function of ever denser cities. It's clear that we can stuff an immense amount of humans into a single square kilometer, but there has to be a downside. That downside is the fact that demand will usually be much higher than supply. If that wasn't the case, we couldn't convince so many people to live so close to each other in the first place.