I don't think this article makes a logical argument at all. It says house ownership is bad because it gives people a vested interest in constraining the supply of housing. That's true but that's not a problem. If people didn't own houses then someone would, and that someone is still incentivized to constrain the supply. I'm also not sure that reducing home ownership is a good idea - "let's make everyone poorer" doesn…
that does not follow - if that someone does not want to sell them or make profit on the rent, why would he constrain the supply?
"What's the actual alternative to people owning their own homes."
The state, the city, not-for-profit entities owning them, for example.
"I also take issue with this idea that we should all be living in skyscrapers."
I agree with that and also think it's much better to live in about 5-story high buildings; the quality of life and price is much better with those.