Yes, that is more in line with Webb's thesis, however, I think you would agree that an Arabic identity began to be (if you agree with Webb) constructed due to Mohammad's rise during his lifetime to an extent, even if it wasn't necessarily "complete" until the Ummayads set about writing their histories and poetry.
Personally, not being a Middle Eastern specialist, it seems Webb goes too far, which seems to the academic consensus. I also think we place far too much emphasis on specific labels, that identities can remain reasonably static despite changing labels, I see this in other areas of history. It seems to me that there was, essentially, a Pre-Islamic identity, Ma'add, which essentially can be called a proto-Arabic identity. Just because such labels, either Arab or Ma'add, aren't considered significant to the people they apply to, doesn't mean they didn't exist in some form in their consciousness or in real terms regardless of their perceptions. That's not to say that what can be called Arabic in the 9th century is likewise proto-Arabic/Ma'add in the 6th century. Interestingly the Lakhmids and Ghassanids generally seem to be considered outside of the proto-Arabic identity due to their Imperial associations. I wonder if, in non-Islamic Arabic conquests, how religious differences will influence the establishment of Arabic realms, which ones will be Monophysite or Orthodox, and which ones will be Nestorian, etc. I think the Sasanians are in deep sh*t after losing to Heraclius, but I don't know if the Byzantines will fall so lackadaisically to a non-unified Arab army.