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.
The process of getting a license plate once home was actually a breeze. I used the website of the Dutch Vehicle Authority to make an appointment (for the following day at 2pm) and they gave me a temporary license plate. I simply had to write this on a piece of cardboard and put it where the license plate would go. Called the cheapest insurance company to get temporary insurance, which was no problem, and simply drove the car to get it inspected.
If you buy a car (which is currently unregistered) from a private person that doesn't have those, you just did something that the bureaucracy didn't foresee and at that point, the best course of action is to avoid as much of it as possible.
At this point I have no hope that any of this will getter unless the german state collapses completely.