See https://de.wikipedia.org/wiki/Impressumspflicht#Telemedienge...
Source: I'm a German, I publish websites.
Source: I am also a German, I publish .com websites.
Interestingly, you only have to do it IF you have an online presence. As in, you don't have to have an online presence when running a business.
I think the point here is to not disclose your identity online as a defense against bullshit lawsuits. Whether you agree with the approach itself is one thing, but if it works at protecting against lawsuits it'll definitely work at protecting against enforcement of this law.
Furthermore, when it comes to the lack of "impressum", merely determining whether the law has been broken would be difficult with proper anonymity. Let's say he's based in Germany but anonymously purchases hosting in the US or another country and never discloses his German residency on the website - in this case it's impossible from the outside to even determine whether he is actually breaking this law, removing the probable cause for an investigation in the first place. As far as I'm aware, Germany does not - and wouldn't have the resources anyway - to thoroughly investigate every foreign-hosted website to determine whether its owner is actually based in Germany and is thus breaking the law by not publishing an "impressum".
Honestly, if the world was run by such people, we would be still in discussions and evaluations.
Sometimes you just have to accept facts and move on.
I remember "the good old web" (tm) where people did stuff anonymosly. In this way people can stay safe and continue publishing stuff.