"but it does indicate that an abundance of cheap housing/buildings makes culture thrive."

Not on its own, though. Plenty of abandoned/underpopulated cheap places in europe that do not thrive. But it certainly is beneficial.

(in the case of Berlin, there was for example a special effect, that all germans living in west berlin did not had to go to the army (to not having to shoot their relatives in east berlin) - so lots of counterculture people evading the army came to Berlin and they created culture)

It's politics that prevents the construction of cheap housing, not capitalism.

Don't you think that a lot of that politics stems from politicians wanting the value of houses in places where they (or their friends) live to go up rather than down though?

Capitalism is a subset of politics

But capitalism doesn't have an interest in prohibiting housing being built in certain areas or limiting density to a fraction of what was possible 130 years ago.

It doesn't? People owning housing has an interest in keeping housing supply low to make the value of their assets go up.