Presumably this means industry in general, rather than necessarily a specific industry, so "Industry" without the article should be allowed.
Is singular βιομηχανία used in this generic way in Greek? E.g. would one say "Η ελληνική βιομηχανία" to refer to all the industrial activity in Greece?
Yes, that correct.
Is Bohuslav1 correct? Why is "the" required here?
…not "life-hack" ;)
Why not "the factory"?
Factory=εργοστάσιο (the building). Industry is more like the process of producing a product.
Is "βιο" supposed to reflect a root in the word for "life"?
Just search for βιομηχανία on https://en.wiktionary.org/wiki/