I do like living here, though I wish the school day for the state-run bilingual schools were not so long! (8 till 4).
I like the fact of being surrounded by visible history.
You end up HAVING to talk a lot with your children about WW2, Nazism, East/West Germany and so on because so much of it's legacy is still present. E.g., remains of the Berlin Wall are there, East Berlin buildings still show WW2 bomb damage, as do statues and so on. There are Stolpersteinen all over the place giving the names of deportees to concentration camps. This kind of thing.
On top of that there is such an access to the forest and lakes around Berlin. In more urban areas, there are also a HUGE number of playgrounds, especially in the more central parts of what was West Berlin. I once did a mental counting up of playgrounds within a 20 minute walking distance of our apartment, and reached 25.
There is a also a lot available for kids, if one is so inclined, such as opera for children, activities for children in museums, the technical museum...
It is definitely expected that parents are more hands off (though you will be getting a bigger range of parenting styles at a a bilingual school). Yes, children do play a lot more alone at the playgrounds, and it is considered fairly normal that primary age children go alone to school. Although, again, there will be fewer doing that at a bilingual school, as the kids tend to be coming from further away.
Lots of kids are going to organised activities after school and being taken there by their parents, though! As well as the whole birthday party rounds on the weekends.
Greatest challenge for me is actually the general Berlin rudeness. They go on here about "Berliner Schnauze" as if it is a good thing, but personally what I see is people being horrendously rude and inconsiderate to one another, and I don't want my child growing up thinking this is appropriate behaviour! (However, using public transport, I tend to see a lot more of this. I suspect people who don't use it would probably have quite a different opinion and think people are a whole lot nicer.)
Another issue is rigidity and blaming. Almost nobody will EVER admit personal fault for anything, or simply say that a situation is a certain way without any particular person being a cause of it, but instead they will always try to fault the other person, rather than simply trying to work towards a solution.
So e.g. to keep it in the context of schooling, if you were in the situation that your child's German needed to improve to be on par with his/her classmates, the teacher would be more likely to tell you that your child is a hopeless learner, and it is all his or her own fault. Or that you, as the parent, are at fault. This particular example has happened to several parents I know, and needless to say, it is not particularly helpful!