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.