Bürgerämter are most of the time a fucking joke. My registration in Berlin took months after I already moved there, the waiting times are just that long, and this seems to apply to many cities. I live in another city now, and my ID card has been expired for months now (which, legally, is a misdemeanor). There isn't a single free appointment anywhere, citywide. You can attempt to go personally there in the early morning, yet here is what I encountered: arriving half an hour early to the Bürgeramt: THIRTY people waiting there, squatting in the hallways, all the way out to the door. On another day, arriving an HOUR before it opened: 12 people already waiting. It's all a joke. And this isn't a recent phenomenon - it's mismanagement for decades, the people responsible should all be fired (but of course that isn't possible).
There should be a "Minister for Time", who has the authority to crack down on such bullshit, not only in the German state bureaucracy, but also in the medical system (good luck getting any quick care here!). Both have taken to a level that is undignified, and wastes person-years of sitting in depressing places. Waiting should be an exception, not the norm, and there need to be metrics against that which have consequences.
For what its worth, I was pleasantly surprised by the things you can do digitally in Kiel.
Schleswig-Holstein, Niedersachsen, Hamburg and Bremen have "Dataport", a publicly owned company which does a lot of the tech for public institutions in the area. While they are not operating at the efficiency of a normal company, they seem to be fairly useful. I'm not sure if something similar exists for other regions in Germany.
Now this has changed a lot in the last decade or two, but it has been accompanied by avg. yearly net migration of 80k people which is putting a major strain on all public services.
Berlin isn't centralized, but the districts (Bezirke) have quite some freedom to structure their work (similar to "kommunale Selbstverwaltung" in other states) and how they run the administration offices (Bürgerämter etc.) which then leads to city administration passing a law for something and each district implementing it differently and city administration having no authority to override district decisions, while taking the blame.
Fixing this requires a change to the constitution, but that requires a 2/3 majority in the city assembly (Abgeordnetenhaus) or a referendum. Getting the majority is hard as there are varying differences (from parties being happy about number of posts to full, like district mayors etc.) and "dislike" between districts (East/West, center/outer, ..)