If groups A, B and C have no influence on groups D, E and F, then you can have separate ANOVAS. But how would you know about interactions until you test for them? Usually it's best to have one big ANOVA that includes all factors to capture all interactions and to reduce Type 1 errors. I review papers and I would definitely notice :)

Does the omnibus test have any rationale? It tests a hypothesis so vague that I've never used one in a long career. What hypothesis says that there's "some kinda difference"?

Report only on tests that are grounded in your study hypothesis and linked to theory. Otherwise you end up having to explain away results that you didn't need.