Cities have high property prices because they are extremely desirable.
There is scarcity which mean those selling can reduce the set of candidates by pumping up the prices. Its a simple mechanic, not a very sophisticated approach...
There must be a way of determining who has the right to occupy the scarce resource..
yes! exactly! We haven't attempted to examine the question let alone attempt to answer it.
Lets say we have:
1) renters serving small numbers of customers.
2) renters serving large numbers of people.
3) renters rarely serving the same customers.
4) renters regularly serving the same customers.
Lets say (for sake of simplicity) people spend half their income on rent, lets say we can infinitely increase rent as the bill is (inevitably) forwarded to the customer they serve (everything involving labor becomes more expensive) and lets assume perfect scaling.
Then you can (and we do) have a city where wages are low and everything ("produced" locally) is cheap OR you can have a city where everything ("produced" locally) is very expensive and wages are high.
You somewhat increase productivity by forcing out people who can only do manual labor but it doesn't scale nicely. Eventually the buildings only serve as investment vehicles rather than nurturing the cities productivity.
We should wonder where the money goes really. I have a mental picture where eggs are cheap but if you want a fried egg with a slice of bread it is going to cost you 100 times the price of an egg because the fried egg is mostly made up out of rent.
I think it would make much more sense to have government do the wealth extraction and to configure rent to have the proverbial eggs fried cheaply.
Cities not just import stuff produced elsewhere (because they cant do it themselves) but they also export a lot of valuable services. Those would be a lot more attractive if they didn't scale with rent prices 1:1.
I'm not claiming this is the holy grail, its a rather raw train of thought. It seems to me to have as much economic activity as possible one could force the price of the raw resources down by a whole lot. Forcing people to fry the egg at home doesn't make much economic sense, we can have professionals do it for you at scale.
It could be interesting to run the numbers on room service vs having your own kitchen? Why do we need a million kitchens in our city? They are just sitting there taking up space not being used? It seems as bad as having the empty buildings increasing their own value by making the resource more scarce.
One more thought:
Do we want university graduates or do we want land lords? Why would we want land lords to enrich themselves with money the students don't really have? The student automatically becomes an expensive employee. Education turns into a scarce resource. If it was a game (and it is) the "let the market figure it out" approach is easily defeated.