Is David Graeber correct to assert that patriarchy did not exist in Sumeria?

_{This is preliminary, since comments are too limiting for this kind of discussion. Hopefully this answer will be evolving with the question.}

There are quite some intricacies involved in answering this question properly. The first hurdle is about definitions. Graeber states in accompanying endnote:

32. Obviously I am distinguishing the term here from the broader sense of patriarchy used in much feminist literature, of any social system based on male subor­dination of women. Clearly the origins of patriarchy in this broader sense must be sought in a much earlier period of history in both the Mediterranean and Near East.
    _{David Graeber: "Debt: The First 5000 Years", Melville House: New York, 2001, p 415.}

What he exactly means by this "obvious" distinction remains a bit in the double category of unclear unclear, unfortunately. But we might use our own, or lets say commonly accepted definitions in lieu.

The second hurdle is the exact timeframe. Graeber really doesn't give an exact one, in numbers readily translatable or an elaborated fashion illuminating the shifts, transitions and developments. Except he does use this:

In the very earliest Sumerian texts, particularly those from roughly 3000 to 2500 BC, women are everywhere. Early histories not only record the names of numerous female rulers, but make clear that women were well represented among the ranks of doctors, merchants, scribes, and public officials, and generally free to take part in all aspects of public life. One cannot speak of full gender equality: men still outnumbered women in all these areas. Still, one gets the sense of a society not so different than that which prevails in much of the developed world today. Over the course of the next thousand years or so, all this changes.
_{Graeber 2001, p178.}

This clarifies his frame of temporal reference and implicitly it also reveals a bit on what he uses as a definition for "non-patriarchy" –– and emancipates his views from purely feminist ideas about the early matriarchy, ironically.

Two broader definitions for patriarchy might be:

–– social studies: a form of social organization in which fathers or other males control the family, clan, tribe, or larger social unit, or a society organized in this way
–– social studies: Patriarchy is also the control by men, rather than women or both men and women, of most of the power and authority in a society.
_{Cambridge Dictionary: Patriarchy}

Especially in the laxer second variant we see the most common understanding reflected, that patriarchy is 'men are in the sole control of everything, officially, including the women.' Control all the way up to including wives as property:

Traditionally, patriarchy gave the father of the family complete possession over the spouse or wives, children, etc. as well as the ability to perform physical exploitation and every so often even those of manslaughter and auction
_{Wikipedia: Patriarchy}

If we look at the rough timeframe for Sumeria we might use this excellent summary:

In ancient Sumer – now modern-day southern Iraq – women enjoyed the same privileges as men in both society and commerce. But when the Akkadian King Sargon conquered, and Sumer became a Vassall state, the outlook for women drastically changed.
The patriarchy promoted by law
When civilisations begin to write down their laws, this is when the patriarchy becomes enshrined. There is a phrase on the Enmetena and Urukagina cones – the earliest known law codes from circa 2400 BC – that says “If a woman speaks out of turn, then her teeth will be smashed by a brick.”
Later, the Code of Hammurabi (circa 1754 BC) of ancient Mesopotamia proved a mixed blessing for women. The laws recognised the right for women to own property, while also forbidding arbitrary ill-treatment or neglect. In widowhood, wives were allowed to use their husband’s estates for their lifetime.
However, the code was a blow to women’s sexual freedom. Husbands and fathers now owned the sexual reproduction of their wives and daughters. This meant that women could be put to death for adultery and that virginity was now a condition for marriage.
_{BBC: The Ascent of Woman}

As this is indeed the current standard:

WOMEN. There is documentary, visual, and archaeological evidence for the role women played in Mesopotamian society through the ages. In many early textual sources, however, the gender of persons mentioned is not always clear. It appears that in the Uruk period there was, at least ritually, a complementarity between male and female; the highest male office (EN) had a female equivalent (NIN), and both are depicted as officiating side by side at important functions. During the Early Dynastic period, women could also occupy highly prestigious offices, as the grave goods in the “Royal Tombs” at Ur and inscribed votive gifts demonstrate. According to the Sumerian King List, there was even a female ruler of Kish.
It seems, though, that female status at high levels diminished progressively after the Early Dynastic period. There were some remnants of influential positions, such as that of the entu priestess of the moon god at Ur, which was often held by daughters of the ruling king. Princesses and queens owed their social rank to their relationship with the king and especially some queens could at times hold the balance of power after their husband’s death . Royal daughters, on the other hand, could be married off to secure political alliances and to provide an informal intelligence system.
Written documents also shed some light on the legal position of women in Mesopotamia. They could hold and acquire property, slaves, and other valuables; invest their dowries as appropriate; engage in business ventures of various kinds; and begin litigation.
_{Gwendolyn Leick: "Historical Dictionary of Mesopotamia", Historical Dictionaries of Ancient Civilizations and Historical Eras, No. 26, The Scarecrow Press: Lanham, Toronto, ²2010, p 189.}

Conclusion: Graeber is largely correct in emphasising that the general social status of women in the region was much more equal than in the thoroughly patriarchal now, that due to scriptural references dominating the perceived history for a long time was thought to be of almost eternal rightousness at face value. A difference from patriarchy is not automatically the opposite, in this case matriarchy. What he means, hopefully, is not the reign of women, but much more egalitarian equality than expected.

An interesting summary about the current scholarship concerning The Role of Women in Ancient Sumer was written by Laura Valeri in 2013, sadly without further references.
^{Thanks to @J Asia for finding this and alerting me to it.}