> But all landlords would pay the tax, and none of them would want to lower prices below their own costs.
This is true ...
> If it becomes impossible to rent properties profitably, then eventually there would just be fewer rentals available.
But this is a leap that doesn't apply to the Georgist scheme. A key feature of the scheme is that taxes on buildings (possibly among other taxes) are removed in favor of the land value tax. It's true that if the tax on some piece of land increases sufficiently, it will eventually become impossible to profitably charge rents on existing buildings on that land--but improving existing buildings or building some superior building (for example, a new building with more units than a pre-existing building) allows landlords to continue profiting.
If the current landlords don't want to improve buildings there, they can sell the land to someone who will. And it's clear that someone will, because if there's enough demand for the land that the land value tax makes it unprofitable to sit on it without improving it, someone is going to want to do something more with it.
The reason we don't already have a Georgist-style economy is quite obvious, of course: the homo sapiens landlordicus is a very soft, squishy, vulnerable subspecies. If they had incentives to improve their properties and thereby possibly even work as hard as the rest of the human species, they might well go extinct. And no one wants that.
That would require removing the zoning restrictions that prevent the construction of new housing.
But here's the rub: if you remove those restrictions, you don't need to change the taxes. Even with current taxes, people will build more housing units, if they're allowed to and they can make money from it.
If it becomes impossible to rent properties profitably, then eventually there would just be fewer rentals available.