There are a lot of people to rent from that don't use that money to directly pay their mortgage.

How does that relate to your assertion that people buying a place and renting it out do not add any value?

There are a lot of places to rent out that are already outright owned, and there are a lot of places where buildings can be built to be rented out, thus increasing supply.

People getting a mortgage and renting out the mortgage adds systemic risk without increasing supply, it doesn't provide any service that isn't provided otherwise.

If a building is built to be rented out, does the builder typically rent out the place? No, it will be someone who buys the building, typically with large amounts of borrowed money, who will then rent out the place. Builders build (often also with borrowed money) because they can then sell the building to someone who can buy. This is all part of the economy.

And you're certainly misusing the term "systemic risk" here - a small-time landlord borrowing money to buy a rental does not in any sense contribute to any kind of systemic risk. They are taking on personal risk in the hope of generating a return on their investment.

The case that was mentioned in the OP is of someone who is renting out only one floor that they bought.

Besides, the main cost in building in NYC is land prices - in the long term the builder isn't worse off by decreased land prices, and neither are renters, while people who want to buy a house are much ahead.

I have no idea what your point is.

You could make the same point for just about everything. Starting a restaurant doesn't help anyone because you're just buying food ingredients, labor and real estate and turning that input into food and anyone can do that so you're not increasing the supply of food. Investing money in the stock market certainly doesn't help anyone since you're just buying existing securities that were already issued. The economy as a whole is full of things that aren't that important independently that you could make a convoluted argument against, even those things often add up to a functioning economy with a robust system for price discovery.

In Vienna, you could rent from a city-owned enterprise.

Why does Vienna provide this service if it adds no value?

Vienna built a bunch of social housing under social-democratic rule to decrease rents, and unlike most other cities, they opened it to a relatively wide income bracket. The value added, if you want to call it that, is _less_ money is spent on rents in the city. Supposedly, a majority of units in Vienna still falls under social-housing restrictions, though I don't have the time to dig deeper now and so I can't tell you what proportion is directly owned by a public enterprise and what proportion is housing co-ops (a somewhat popular model in Europe).