That's not true at all. The number of building permits granted has actually dropped and when new developments are granted, the number of apartments build will always be negotiated down from the original proposal.
It's insane. The dense Gründerzeit quarters are the ones that are most popular, yet we are unable to build anything alike.
Ok. But the real statistic to disprove the statement is the percentage of building permits granted based on building permit applications. Has that dropped? Or is that still rather high? Also, I think you're not mentioning the fact that major parts of Berlin are against gentrification so while the local government want something, the people don't, because that gentrification allowed them to bypass the rent rise law.
Realistically, to say they aren't building a ton of new housing at a rather fast rate would be a lie. As fast as possible, maybe hyperbolic but that is all.