Speaking from a London perspective, though I'm sure a NY one is worse, not better.

> Sry lack of new houses seems doesn't seem like a cause for homelessness. If you can't afford a house in the big city, you move to a smaller one.

This would only make sense if the smaller houses were reasonably priced as opposed to the bigger ones. This is not the case.

And creating more housing would absolutely be a step in the right direction in terms of reducing extreme housing prices. Unless you don't believe in demand and supply economics, that is.

> Once they're down, it's almost impossible to get back up

Yes but that's partly because they can't afford to rent even basic lodging, let alone afford to buy one, and a basic roof over one's head is a pivotal basic need for most things one needs to do in life.

This kind of problem rarely has a single cause but housing costs put pressure on many of them. High housing costs mean people save less and are more vulnerable to other events cascading into homelessness. They also increase stress levels because people live further away from their jobs and spend more time and money commuting while being less able to care for their families, which can have generational impacts when older kids aren’t studying because they’re taking care of younger siblings while their parents work and commute. It also inflated prices for almost everything else because businesses are paying more for their space and have to pay their workers more, too.

That doesn’t mean that we don’t also need things like better support and easily-accessible government healthcare, but we have to recognize that these things are all connected. Salt Lake City somewhat famously found immediately housing people helps with mental/substance abuse issues simply because all of the other problems in life are more approachable when you’re not sleeping on the street, missing appointments, and having your essentials stolen.