Increasing supply outside of them, which would be easy because Berlin is surrounded by land you could build on won't change that. At least not without billions of investments in public transport infrastructure, maybe.
Any regulation like building requirements is probably federal, I doubt Berlin has any extra regulation. So there is nothing Berlin can do here.
Removing regulation that is too tenant-friendly isn't popular with voters, it's not going to increase the supply of apartments for students and poor people. This kind of regulation including but not limited to the Mietpreisbremse is also federal law and not something Berlin can do all that much about.
Even central Berlin is full of brown-field sites that could be developed -- Flughafen Tempelhof most prominently. If you go for Hong Kong style density there, a lot of housing problems would be solved ... Laws can easily be changed if there is political will at the local, regional or federal levels.
However, all this requires a bit of long-term thinking and planning, of which there is little evidence. Instead we get populist measures like "Mietpreisbremse" which hinder new developments.
Flughafen Tempelhof can't be developed, voters have voted directly against it. That law could be changed but to do so would arguably be undemocratic and it would certainly be political suicide for the forseeable future. This also brings up the important point of city planning, cities are more than just housing. You need to take that into consideration.
Also indeed increasing supply is about long-term thinking. You can't increase supply very fast and in this instance you can't increase it fast enough. This means you need something to slow down price increases in the short-term which measures like the Mietpreisbremse or banning AirBnB do.
Why exactly should they? I mean, I don't want to forcibly remove them, but there are many cities in Germany other than Berlin.
If one city is getting richer and richer, then the average price of an apartment goes up. More rich people start comming, because they see oportinity to do business there. At the same time poor people have a choice of moving to smaller, more affordable cities.
The biggest pitfall of a mankind is that we cannot grasp that the world is constantly changing. Life itself is a process of constant change. Just because your parents lived in Berlin and you were born in Berlin does not imply that things have to stay this way forever.
Trying to stop the flow of time and inevitable change is what makes us miserable.
Why hasn't anyone thought of this before!
Get rid of the restrictions on height, and build 50 - 100 floors Hong Kong style, at least in the center.
Tempelhof shows the stupidity of the Berlin voter. The refusal to develop even a small part of the airfield was almost exclusively justified with fears of "gentrification". Congratulation voters, you have just voted yourself into higher rents. (Also, one could have a new "Volksbefragung", in the light of the dire need for more housing in Berlin.)
people with low incomes should be
able to live in the city.