The problem is, that only works if your Aufenthaltsbehörde isn't swamped in case load. Unfortunately, the ones in cities where tech workers are wanted are swamped and often times you need a lawyer to file an Untätigkeitsklage (inactivity lawsuit) or threaten to do so to get them to respond.

The Ausländerbehörden are massively understaffed (well below 50% of what would be needed), and work distribution usually is that anything attached with a court deadline has absolute priority, anything from a lawyer comes next, and whatever comes from a generic person or company just gets shifted to "Ablage P" (the paper recycling bin).

You're right, but I have to say that the Berlin immigration office has greatly improved in the last two years. The wait times are getting a bit more acceptable.

The biggest challenge now is housing those newcomers. The housing situation is turning people away.