I asked

> In such a scenario, where would the Sumerians have come from?

Heike replied

> Actually, the "e pluribus unum" idea they might have been present
> in the area since long before has a similarly structured problem,
> namely identifying which advantage a bunch of hunter-gatherers from
> the marches should have taken over a neolithic culture?

Given that the East Arabian Upper Paleolithic stretched from Kuwait
down to Abu Dharbi, and it was this hunter-gatherer group that was
acculturated to Neolithic ways by the Ubaid culture in Qatar and
Bahrein, I suspect that these were the people who later became the
Sumerians. This is in part confirmed by R. Macchiarelli in 1989
"Prehistoric 'Fish-Eaters' Along the Eastern Arabian Coasts : Dental
Variation, Morphology, and Oral Health in the Ra's al-Hamra
Community" where he shows that these people had a dental morphology
that contributed to the Sumerian strata from Ubaid III onwards. The
H3 mollusc shell midden in Kuwait, dates from 6th millennium BCE down
to Early Dynastic times.

Hope this helps