Surely the USA varies a lot. New York may be a close approximation. What's that like?

Not necessarily. In fact, I went to high school in NYC and I distinctly remember my 10th grade History teacher reminding us repeatedly "New York city is nothing like the rest of America!"

I left the city decades ago: he's right!

Yes, owners definitely rent out rooms in New York apartments.

Or hang a sheet across the living room and rent out the new extra "room."

I have actually done this. Well, it was a heavy curtain.

Ah, another city with a permanent housing crisis!

nope, with too many people :)

If it persists, it's the same thing really.