This is just factually not true. There is plenty of empty homes, apartments and condos to house everyone. There are many units kept empty rather than lower prices. It is a myth that supply and demand sets prices.
There is also the effect of rich people needing a place to stash money and not wanting to bother with tennants
Housing activists in Melbourne have been campaigning about that for years now
I did not believe that was true until I stumbled on it happening. A rich lawyer, a salary far too big to spend, so they collected (empty) houses. They did not care about cash flow
A tax on empty houses in areas with accommodation shortage at worse cannot hurt and at best could free up a lot of accommodation