I came across this paper from 1943 by Thorklid Jacobsen that argues that Ancient Mesopotamia was one of the first examples of a democratic form of governance, with an assembly choosing their leaders, and the leaders having to present their proposals and actions before the assembly of citizens for approval. He also argues that evidence for this form of democratic governance continued even after kings consolidated their power in the judicial systems, where common citizens had a powerful voice in how they were judged.
He uses the Epic of Gilgamesh, the Mesopotamian Creation myths and other evidence to reach this conclusion.
Has this theory survived as a plausible explanation of early Mesopotamian governance or has it been abandoned as incorrect?
The idea of "Primitive Democracy" in ancient Mesopotamia seems to still hold currency. Jacobsen's work has been cited and elaborated in books by several big-name assyriologists like Saggs, Oppenheim, and Kramer. I found this conference paper by one Benjamin Isakhan which gives a pretty thorough review of the relevant material; he echoes much of Jacobsen's work and provides quite a few additional substantiating examples from archaeological research, such as the excavations at Ebla (which have revealed "a sophisticated political culture involving some 11,000 public servants" and kings who were "...elected for a seven-year term and shared power with a council of elders;" see Manglapus, 2004), the general assembly in ~2300 BCE at Kish to elect a king (who took the throne-name Iphur-kish; 'Kish assembled'), and the political power of the temple community of Early Dynastic Lagash, which H. Frankfort characterized as "...show[ing] a strongly democratic character". Moving to the 2nd millennium BCE, he also brings up the assembly of Nippur which persisted "well into the Ur III period (~2150-2000 BCE)" and the bicameral assembly of Sippar that lasted a good 300 years, from 1890-1590 BCE, and which retained judicial authority even after being suborned by the Babylonian empire and having much of its upper house "infiltrated by royally-appointed officials" (see Leick, 2001, Mesopotamia: The Invention of the City). Isakhan's paper goes on a bit and cites what looks like a very interesting article dealing with the 1st millennium BCE (Dandamayev, M. 1995. Babylonian Popular Assemblies in the First Millennium) but I can't find a copy online, and that's a bit beyond the scope of your question anyways.
There is a somewhat alternative model of early Mesopotamian statecraft to be found in the work of some Marxist assyriologists, specifically thinking of I. M. Diakonoff. He seemed to generally be in agreement with the basic idea of primitive collective governance in early periods of Sumerian history, but focused more on how that tendency waned with the advent of despotic states. There were some interesting points raised in his paper/book chapter The Rise of the Despotic State in Ancient Mesopotamia that I'd like to quote, but I'll have to wait until I can find a copy of it again.
But yeah, definitely check out Isakhan's paper, the bibliography is chock-full of pertinent stuff.