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 alo

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).