I don't believe that is the case. There is no value in letting a property sit empty accumulating value, when it could be doing exactly that while pulling in a hefty monthly rent.
It may well be the case that there's enough homes empty to house everyone, but only if they're happy to move somewhere they don't want to be, and where there's no jobs to pay for them.
For context, I live in greater Boston. The state here has actually forced towns and cities in the inner belt around Boston to remove their zoning laws anywhere within a mile of a train or subway stop. Thousands of additional condos and apartments have been (and are being) built as a side effect and we will have dozens of new squares with shops and businesses on the ground floor and tens of thousands of new residences surrounding our transit hubs.
It’s not been smooth. Some towns are suing the state and doing other random bullshit to slow the process. Pushing these rules to the federal level would actually help states and metro areas consisting of multiple connected cities address the issues more efficiently.