They're pretty independent, but maybe not independent enough for some? I mean it's no issue concerning Deutsche Welle, they are specifically non-independent as an institution of German culture politics. But the public broadcasters have supervisory bodies that are composed by law and contain

representatives of socially relevant groups, such as political parties, trade unions, social organizations, churches, etc., as defined in the broadcasting laws of the federal states [...] whereby the political parties - limited by the courts - do not hold more than one third of the seats.

This does not take into account how the representatives of the christian churches always vote with the christian parties and vice versa, the atheist representative is also a member of a political party and many social organizations depend in their work on the goodwill of the government.

Imho. that's not a problem, politics touches everything so you'll always find political connections of some kind everywhere. But seeing as our neonazi party is gaining momentum, maybe we should think about further reducing political influence, so it doesn't fall into their hands in a worst case scenario. Or we could reduce religious influence, following recent waves of Germans leaving their churches.