It's both true and untrue.

It's true that finding appointments can be difficult if you just open the website once and see a calendar with no available slots for months; _but_; there are very frequent cancellations and some offices releases slots for the same day at midnight/7AM/etc; so if you're determined and flexible; it's quite doable to get an appointment ~soon.

For some things (apartment registration) you can also just show up on the day and just wait.

(My last interaction with a Bürgeramt was in Berlin in October 2023 though and was pretty similar for the previous ~8 years; things could have changed in the meantime!)