I think this article points to a much more important issue. And that is existing real estate owners acting to keep the value of their holdings as high as possible, i.e., to keep prices as high as possible. A lot of the wealth in America is locked up in real estate and this is especially the case in large cities. The landowner lobby (although they do not call themselves that) is quite powerful.
Actually the fact that rent control comes up so often as an issue is just another piece of evidence that the landowner lobby is so powerful that they are able to completely control the conversation. When faced with what is basically an accusation that they are gouging the population they turn it around and declare that the few weak laws that slightly reduce their ability to gouge are really the problem.
And while rent control is a point of contention, the real way the landowner lobby keeps prices high is by restricting new construction. This is a much more important factor than rent control.
A very important issue here is that homeowners and landlords are much more active in most political processes. If one person decides that he will dedicate his life on ownership of land and will make that the main source of his income, he will easily see that it is important to devote time and energy in a lot of the rather tedious local political processes. If a second person decides to dedicate his life to some type of profession or art instead, then local politics tend to become a bore and a time waster. But then the second person is not at the table when important decisions are made and his interests are negatively impacted.
So the general answer is to be more active in politics. But how to be more active in politics?
Here is an issue that the article did not really touch upon. The most important requirement for more construction and higher density living is the availability of infrastructure. Nobody will approve high density buildings without the infrastructure, nobody wants to create Mumbai style tenement cities.
And the most important and most expensive part of infrastructure is transportation. Look at any successful large city in the world -- they are usually defined by their subway system. After all the subway is the only sane way to move large number of people around in a very dense environment.
In fact the main battle against development of LA has been fought for the last 20-30 years over the LA subway. There have been some very successful attempts to slow down and block the development of the subway launched mostly by neighborhood groups. As a result the subway came in much later than originally planned and the most important and most desired line (the purple line) is still only a tiny fraction of what it should be.
So if you want to help create more housing it is always best to vote in favor of new transportation projects, especially high density high efficiency projects, like the subway.