The wait time can be shortened on many factors, but people search for specific homes. There is a shorter wait on 2 bedroom apartments, while longer on 3 and 4 which is why families get bumped up the list on 3 and 4 bedroom apartments, 5 and 6 are short wait too but too expensive to live in for the average city family. Then it is down to location. If you wish to live in a big city, then you need to wait longer for you can't just build new homes there, if you wish to live in the outskirts, shorter wait and so on.
Denmark's homeless population is VERY small compared to our population, and a grain of sand compared to the US.
In 2019, there were 5.8 million people living in Denmark, out of that number, only 6.431 were homeless for various reasons. These numbers consist of people who have chosen to not be part of the main whole or chosen to be 'free'. You can't solve problems that do not wish to be solved though. And if these wish to be part of the society, they can go to the local office of where they are currently and request help to be housed and request benefits. They hardly take spots from the queue from others when there is thousands of housing queue listings.
This statement sounds really uncomfortable. Building new is not always the answer. Especially not when you are in a small country. As well, you have to remember, queues are made for people waiting for homes to be constructed too, or waiting to have money for it, or waiting for the place to be renovated.