They're not doing it willfully. They're doing it because they have no choice. Let's say the current average rent in Seattle is $1,500/month. Why $1,500, and not $2,000/month? Why are the landlords willfully cutting into their profit margins by not increasing rent? We both know the answer: because the current price is based on what the market can bear; in other words, it's based on the relationship between supply and demand. They can't arbitrarily decide to raise rents without risking losing tenants and having units go unoccupied.
Now, let's say the property tax increases a bit. Will that be passed onto the tenants? Well, can the market bear it? If no, then it won't be passed on. If yes, then why wasn't the rent increased before?
Like I said before, it's different in low cost of living areas. There, the profit margin on top of the costs is relatively small. The rents in that case aren't really based on demand, because supply in those areas tends to be very flexible, you can always make more housing to match demand. For a property tax increase in those areas, the costs will be passed onto the renters, because the primary determinant of rental prices is cost.
This is why house prices are just as insane in Silicon Valley as rents.
He's claiming that a land tax would not be passed on to renters if instituted, because rent is being driven demand-side (what renters are willing to pay) not supply-side (cost to provide a house to people). Any attempt to increase rent would result in a lower demand by renters, meaning the market would fail to clear -- the rent would then need to be re-lowered.
But you are correct, that rent shouldn't be higher than purchase plus upkeep. In the event of a land tax, rent holds constant but upkeep increases (taxes); consequently, for the equation to hold, purchases prices (and thus land values) will fall.
And that's why this is so politically impossible to pass, even though it is considered the "least bad" tax economically. Land owners would see their assets sufficiently fall in value if a land tax were enacted.
Rents do go up when wages go up (SF) and when there is speculation they rise but it's capped by wages.
This is only true if the rent is based on the cost of the unit in terms of purchase price and upkeep. That's true in lower cost-of-living areas, but in expensive areas, the rent is high primarily because demand is outstripping supply, not because the cost of landlording has gone up. And if rent prices are not based on cost, then an increase in cost (which is what a property tax increase is) will mostly not be passed onto renters.