If English, German, Swedish, etc. shared a single prestige form, which all educated speakers understand and use for formal purposes, I think it would be reasonable and common to call them dialects of "Germanic".

More generally, you'll drive yourself crazy trying to form an opinion of what a language "really" is independent of what the people who use it believe. Norwegian people can generally read Danish, and Danish people can generally read Norwegian if it's written in Bokmål, but Bokmål is only one of the two equally valid ways to write Norwegian. What's going on there? You can trample over the cultural context to impose some abstract definition of what's "really" a language, or you can accept that Norwegian is a language because Norwegian speakers say so and not worry too much about the metaphysical status of Nynorsk.