There is no "problems with solutions", like they say so often on the left, only trade-offs. Someone will benefit, and someone will pay the cost of that benefit.
That doesn't necessarily mean that it's always wrong to do so. Only that both sides of the question should be considered.
And I'm very pleased that the question has been raised here.
As for me, I say displace. The costs to fixing housing pricing are much higher in terms of reduced quality of life, than anyone has addressed here.
Apart from those undocumented costs, there are decades of data available to how fixed housing costs raised overall housing costs and reduced availability in New York. I can personally remember acres of abandoned buildings in Queens (which by the way are all re-occupied since rent controls have been grandfathered away) because the landlord couldn't justify the costs of occupation.
And in spite of the nonsense said earlier, there is not an endless supply of people willing to pay X to live in the south Bronx. They will only pay X for housing which meets their expectations of the value of X, so supply and demand will (of course) work in New York, as it does everywhere. Manhattan is an island, and Staten Island obviously is an island. Even Brooklyn and Queens are on the Island of "Long Island". But the problem of limited floorspace was solved with the invention of the elevator, so supply of housing has continued to expand in New York City, even though the space available has (for the most part) not.
But the argument over fixed housing costs is moot. It has been settled by history. And history has sided against Keynes, and with Hayek.
Government economic controls don't work the way they are intended. And that obvious if you judge them by their result instead of their intent.