There are over 2mil dwellings in the Sydney area, this is not 1/3rd, it's like 8%-9%.

Build some more hotels and houses, this is literally Australia's megacity, it's their New York, of course there's huge demand for both permanent and temporary/tourist housing. People will eventually migrate to other areas if the city doesn't figure out how to build enough, and that's fine, too. Migration to and from cities is normal and healthy. New York City hasn't had population growth in like 60 years or some shit because it can't expand much more and isn't a very economically desirable place to live for a lot of people as a result, to say nothing of the midwest cities that have all been seeing their populations drop since the mid 20th century.

If a city has enormous demand, build more supply. That's literally your only solution. Rent controls don't work, it's been tried and tried and tried again. Your one and only solution is to build more. It will take time and people will have to leave the city if they can't afford to stay there. That is normal and that is just what happens. Sydney will learn to cope. Cities don't infinitely grow, sometimes they shrink or stagnate while they figure themselves out.