We actually have had something not completely dissimilar to this since the 1980s. Anyone who lives in state housing for at least a minimum period of time is able to buy that home at a subsidized price. As local authorities are banned from using this money to build more state housing, this leads to long term shortages in state housing which is one of the causes of the problems we are facing now. (Source: https://en.wikipedia.org/wiki/Right_to_Buy)
Nobody is expecting to get a cheap home in Kensington or the equivalent super-expensive part of whatever city you are from. But it is not just the "cool", "nice" places that are becoming unaffordable. And the problem is that people are not paying to live in these homes. People are paying to use them as big, shiny piggy banks in the sky that happen to offer a better rate of return than other investments. Poor people who actually live in London and use London businesses generally contribute more to London's culture than rich people who don't.
How much would you say contributing to the culture should be worth?
And I still don't understand how they make money by letting houses remain vacant?
It kind of sucks, but what would actually be a better system? That people get assigned places to live by lottery, so if you happen to live in a cool place on the cheap, you were simply lucky? It sounds good on the surface because you only have the people living there in your mind. You forget about all the other people who never get a chance of living there. Why is it fair if some randomly selected people get to live in nice places and others don't?
I am not convinced that the market approach is not preferable. After all, those people PAY for the privilege to live in a cool place. If they have the money, presumably they should have delivered something good to the world in exchange.
And, hello, welcome to overpopulation. The number of nice places to live is finite. Even if we can cram ever more and more people into the world (by building skyscrapers or whatever), it doesn't seem obvious to me that we can provide nice places for everyone. For example, the length of shorelines is finite (glossing over fractal issues), so if your idea of living in a nice place is to live by the sea, there are only so many people who can have that privilege.