Cities with higher densities exist and they're not as nice as suburbia, but it's much better than no housing at all.
The highly parts of existing cities are limited, but as history shows - you build new cities. Population has moved from one place to another for centuries and new cities sprang all over the world. If this was done before it should be done now. Overregulation may be the culprit.
The main "successful" new cities these days are when governments create one for their headquarters (or move their headquarters to an existing smaller city), like Sejong in South Korea. This is because it induces demand quite successfully for government employees to move there. You need a strong inducing of demand to create a new city; existing cities already have strong demand via jobs, other people, and services, so new cities need to outcompete them.