This isn't just about the welfare of current renters. The city (and the people in these quarters) are also concerned about the changes a neighbourhood experiences when the population changes.

What often happens is that these cheap but beautiful areas attract students/artists/etc, who create (or support) lots of businesses and events that make the area attractive. When the lawyers/doctors/etc replace them, they destroy exactly what it was that attracted them to it in the first place.

(I live about 300m from the building discussed in the article)