> If landlords become unwilling to pay for new construction, that reduces the demand for new housing supply hitting the market. Which reduces its price and makes it more affordable to owner-occupants.

Construction occurs until the point that it stops being profitable, which is a price point more than a number of units. Zoning rules make it so you have to buy a 5-story building, then knock it down and pay the cost to build a 10-story building in order to add 5-stories worth of units. That's much more expensive than putting a 5-story building where there is currently a single-family home or an empty lot, which is prohibited. So if landlords stop being willing to pay for that, it doesn't cost less, it just happens less, because at any lower price the construction doesn't happen.

Which is how rents increase. Suppose 100 new units would have been built and landlords would have bought half. They stop buying, so construction companies only build 50 new units because there are only 50 buyers at a profitable price. Then demand for rental properties increases over time but supply of rental units doesn't, so rents go up. Then the landlords start commissioning new units again, because rents have increased enough to make it profitable again, but only for as long as they stay that high. The price of the building stays the same -- it's set by the construction cost -- but the rents go up because the landlord has to recover the cost of the building and the tax, or they're not commissioning a new building.

> Tax what you want less of, right?

The trouble is you don't really want fewer rental units, unless you like high rents. What you want is maybe more owner-occupied units, but this is only a trade off when supply is artificially constrained -- otherwise you can build more of both, which is obviously better.

Whereas building fewer rental units without building any more owner-occupied units is quite useless.