Once you create a list of things that can and cannot be said you have that list on hand. You might have the best intentions about what's on that list but it's better to not have that list at all because it's completely subjective what should and should not be on that list.
It was a local politician abusing his power who is being investigated for that power. It has nothing to do with banning hateful imagery.
Also, I think many people, Americans in particular, overestimate the ability of "market systems" (i.e. "the marketplace of ideas) to self-regulate or the desirability of the results of that self-regulation.
It's one thing to let Nazis march down the street in Illinois, where they are a small group of pariahs with no hope of power; it's quite another to let them march down a street in a place where they've had power and used it to bad ends (and can be assumed to have significant residual support).
While this isn't a specific case of censoring something glorifying or endorsing Nazism directly, they think that their own citizens cannot see such material out of fear that it will inspire people to become militaristic. While the goal of being anti-war and anti-nazi is noble and worthwhile, a lot of things get swept up in the effort to curtail as much of it as possible.
The situation has improved a lot these days after an amendment in 2003. There is an industry board (USK) giving age recommendations (0+, 6+, 12+, 16+, 18+) and if a game has been rated 18+, shops will not sell it to minors. Importantly, if a game has been rated by this board, it can no longer be put onto the index maintained by that Federal Review Board for Media Harmful to Minors. So the far-reaching restrictions against advertisement no longer apply.
From my limited time in Germany, and from my few German friends, I do not believe there is a "high false-positive rate" and they do not "rightfully fear that the system can be abused".
Any system can be abused, but they are not in "fear" of that. They generally support the idea that Nazi ideals need to be curtailed given the atrocities committed to millions of people.
Obviously, and as indicated, this is my very small circle of acquaintances. Hardly a national poll.