> sexagesimal mesopotamian number system was used positionally from the beginning
This is oversimplified. The sexagesimal place-value system developed gradually over centuries+ from various inconsistent unit systems for length, weight, fluid volume, counting, ..., which were eventually (partially) standardized. This long development took place in tandem with the development of cuneiform writing, both ultimately originating in a record-keeping system involving clay tokens sealed in clay envelopes. For a detailed version, Eleanor Robson's book Mathematics in Ancient Iraq is excellent.
I think it's fuzzier than that, both as a historical process (stretching across more than a millennium of development), and as a tool at any particular point in time. Here's some quotation from Robson:
The very earliest literate accounts – from late fourth-millenium Uruk – used commodity-specific metrologies with a variety of different numerical relationships between the units. Those original metrologies continued in use throughout the third milennium and beyond, whether essentially unchanged (for instance areas), or undergoing periodical reform (for instance capacities). They continued to be written with compound signs, which bundled both quantity and unit into a single grapheme, just as the preliterate accounting tokens must have done. Gradually, over the course of the later third millennium, scribes began to write those compound metrological numerals with a cuneiform stylus, rather than impressing a round stylus into the clay in imitation of accounting tokens. Throughout the Sargonic period, and even into the early Ur I period, impressed and incised number notations apear side by side on the same tablets – a phenomenon that has not yet been systematically documented or explained.
[...] But not every new metrological unit was sexagesimally structured. The smaller length measures, first attested in the Early Dynastic IIIb period, divide the rod into 2 reeds or 12 cubits, and the cubit into 30 fingers. None of these newly invented units of measure was recorded with compound metrological numerals, but always written as numbers recorded according to the discrete notation system followed by a separate sign for the metrological unit. This has implications for our understanding of the material culture of early Mesopotamian calculation, as well as for the shifting conceptualisation of number.
[...] At some point early in the Ur III period, the generalised sexagesimal fraction and the generalised unit fraction were productively combined to create a new cognitive tool: the sexagesimal place value system. This calculating device took quantities expressed in traditional metrologies and reconfigured them as sexagesimal multiples or fractions of a base unit, often at convenient meeting point between metrological systems.
[...] The SPVS temporarily changed the status of numbers from properties of real-world objects to independent entities that could be manipulated without regard to absolute value or metrological system. Calculations could thus transform numbers from lengths into areas, or from capacity units of grain into discretely counted recipients of rations, without concern for the objects to which they pertained. Once the calculation was done, the result was expressed in the most appropriate metrological units and thus re-entered the natural world as a concrete quantity.
As far as I understand there wasn't ever a "purely sexagesimal" phase. Some of the units were sexagesimal or sexagesimal-ish and others weren't, all the way along (but gradually trending toward more that were, as various unit systems were reformed). The sexagesimal system (per se) arose after literally centuries of prior experience with some of the unit systems being more or less sexagesimal, accompanying a transition in writing from using marks that were a concrete representation of clay tokens of various shapes toward using cuneiform. Even once there were people using sexagesimal calculations as an intermediate representation, they had to use tables for converting all of the non-sexagesimal units to sexagesimal quantities in terms of a specific unit of each type, and vice versa.
The use of sexagesimal calculation probably had something to do with the use of a type of counting board about which we unfortunately know few details (it was called the "hand" and IIRC could generally support values with 5 sexagesimal places, but anything we know about it comes from scattered textual references and some inferences drawn from calculation mistakes; Robson's book doesn't talk about this subject much but there are some nice papers about it elsewhere).
This is oversimplified. The sexagesimal place-value system developed gradually over centuries+ from various inconsistent unit systems for length, weight, fluid volume, counting, ..., which were eventually (partially) standardized. This long development took place in tandem with the development of cuneiform writing, both ultimately originating in a record-keeping system involving clay tokens sealed in clay envelopes. For a detailed version, Eleanor Robson's book Mathematics in Ancient Iraq is excellent.