We don't even need skyscrapers to relieve the housing shortage. Legalizing duplexes, townhome, quadplexes, condos, or three story buildings that contain six units would add a lot of supply.
What about subjecting this theory to a practical experiment?
In many places of the world, the skyscrapers are full, and the people who move there leave behind another apartment for those who don't want to live in a skyscraper.