Germany has only "soft" rent controls that limit YOY increases in rent and (selectively) increases in rents when changing tenants.
German economic orthodoxy relies very much on the need to build housing in order to deal with housing shortages and with rising rents due to housing shortages. The "soft" rent controls simply buy time to get housing projects underway. Without new housing being built, rents would still reach market levels, it would just take a bit longer.
Berlin's current problem is excessive urbanization driven by migration (1) from the neighbouring comparatively poor East German states and (2) of foreigners who speak only English and hence prefer a city that has already evolved to accommodate English speakers. Various and sundry NIMBY problems where locals oppose new housing projects also don't help.