The backstory here: EU law mandates doing something to reduce pollution if it's too high. German law prohibits various kinds of discrimination. For example, a simple ban on half the cars on Monday and the other on Tuesday would bother large plumber companies much less than independent plumbers who need a carful of tools and have only one car. What actions are the cities permitted to take, and/or required.
I think that's by design. The law is a compromise, written such that the unpleasantness would bite, but would not bite until the people who negotiated the compromise were safely out of office.
True, Stuttgart as the biggest cluster of German car manufacturing has been the epicentre of the German pollution-debate (that was based around particulate matter, not NOX).
The Diesel has become somehow the victim of this (sadly, since as mentioned by other commenters here, Diesel cars have their advantages).
The reason Stuttgart is Germany's pollution capital has almost nothing to do with the Diesel but:
- its topography, being a city almost completely surrounded by hills, thus unusually hot and windless for a German city
- the upper class areas on the hillside/slopes are the main polluters with houses dating from ca 1900 and/or re-equipped with pellet heating systems that were subsidized because of energy footprint while being even worse pollution-wise than even older heaters
Both of these points are validated by the fact, that it's not the weekends when the pollution calms but its the winters when there is almost a daily pollution warning - its when the rich people heat.
Even the plaintiffs in the other recent (particulate matter focused) important German pollution case complained that a Diesel ban does nothing for the pollution situation: The main source for car based emissions aren't the engines - but the wheels.
