Just one more post, 
not as a further argument, but to sum things up and show that this discussion was just about a theoretical difference.
If I understood correctly, Ajahn Brahmali basically argued that ignorance and other necessary conditions are implied to be present throughout the origination sequence of Dependent Origination. With this assumption, individual factors become sufficient conditions for the next. For example, if ignorance is implied to be present in vedanāpaccayā taṇha (“from feeling-paccaya, craving”), then feeling becomes sufficient for craving. But we should also understand this a general sufficiency, not as a universal one, because not all feeling leads inevitably to craving, even before enlightenment. And similarly for some other links.
I don’t disagree with what would logically follow if the assumptions were true. Then we can indeed see each factor as sufficient for the next (although I do still have a slight issue with birth leading to old age, which doesn’t depend upon ignorance and can’t be generalized in the same way).
However, I see things the exact other way around. In my eyes, it’s precisely because other necessary factors are implied that no single factor is ultimately sufficient for the next. And I think Dependent Origination does give universal statements. For example, in vedanāpaccayā taṇhā all craving is dependent upon feeling, not just some of it. This also means that no factor can be seen as sufficient for the next. Otherwise, in this case all feelings would lead to craving. Likewise, all ignorance would then lead to rebirth, even ignorance before enlightenment.
But in practice our interpretations come down to the same thing. We don’t disagree about what Dependent Origination is fundamentally about. The disagreement was just about which logical structure the Buddha used to teach the more fundamental principles that exist in nature.
I hope that sums it up accurately enough. 
If not, Ajahn or others should correct me.