Right, which is exactly why this technology does nothing to solve the problem. The housing shortage in SF isn't because they can't build like Manhattan or Hong Kong there, it's because they choose not to.
Certainly it would be possible to build higher than construction currently goes in SF, but permission is not the only reason why there are fewer skyscrapers. Manhattan and Hong Kong are essentially massive regions of granite that you can build pretty much anything on top of. That sort of geology is not particularly common. You can't just put up skyscrapers everywhere. They'd fall over eventually.
Chicago typically doesn't have well accessible bedrock. I know of one civil engineer who unexpectedly ran into granite and was able to sell it for added profit on a project. Point being, CE's don't typically expect great bedrock yet have developed methods to build skyscrapers in a city that was formerly a swamp.
"He says even though new technology makes it easier to find solid bedrock beneath 100 feet of wet clay, it doesn’t always make sense to drill that deep. Modern engineers still use the same general principle Burnham & Root employed when they floated the foundations of the Monadnock Building on an even flimsier layer of soil known as desiccated crust: They just spread the load."[1]
Look at Manhattan or Hong Kong. You just stack the homes and offices and stores on top of each other.