I'd say all of Germany is culturally very much ok with drinking tap water, Berlin not much more or less than other regions. People who drink bottled water here don't usually do it because they think it's "better" than tap water, it's either because of convenience when you're not home or because they like carbonated water (which is very popular in Germany).
Or because the tap water just doesn't really taste that good. The one coming out of my line just tastes too "metalic", I often tried getting over it but I just can't.
A lot of the buildings in Berlin have very old pipes and while the water to the building is alright, we (in Berlin) have to rely mainly on glass bottles of water (bismark) as if you don’t run the taps enough there’s a nice colour that comes out...
I've lived in a few places in Berlin over the years, I've never seen anything but nice clear water come out of the tap (whether at my place or friend's flats). Granted, it's hard water (lots of calcium) but other than that it's perfectly fine. Pipes as bad as you're describing definitely exist but I'd say they're the exception, not the rule.
I was in Berlin this summer for 2 months (Neukolln), in 3 different airbnbs and the water was not tasty, and was quite bitter. Very different compared to Swedish or Austrian or even Italian (florence) water.
