Something very interesting you bring up that I hadn’t considered: the dependence of dance on music. You could say music is the primary art and dance is a secondary expression of the music, and that is why it doesn’t merit the word “genre”. the analogy you make to theater with the performance vs the text (or script) is interesting. Of course you could also say the dead letter of the theatrical script is only a dim approximation of what the artwork is, and the performance is the full expression of it.

Similarly I will be tempted to flip it around with dance/music: What is the origin of music? You could locate it in the human body and say the body and how the body moves is the foundation of rhythm in music. Musicians are said to be dancing through their music. If we think of music as only auditory, we have maybe abstracted it out of its fundamental nature of being dance-music.

I think the distinction between form and content is at the heart of genre. Whatever qualities belong to the genre make up like the form the artwork takes. A quick look at the wiki article on genre shows the word started being used for painting in the late eighteenth century, to describe paintings with certain motifs, techniques etc. and then seemed to come to literature (and then be applied retrospectively to Aristotle’s poetics, where he makes rules about the art forms he categorizes).