(Pro tip: different cultures have done things differently across time and space!)
I believe that counting method was devised by some fan of duodecimal arithmetic sometime within the past century, and has nothing to do with sexagesimal numeration per se.
The evolution here was: physical clay counters; physical clay counters sealed in a clay envelope; physical clay counters sealed in a clay envelope but also pressed into the outside of the envelope to indicate how many; clay tablet with counters pressed into the outside (since the envelopes with counters inside were redundant); clay tablets with little cuneiform symbols to represent quantities, differing by type of object being counted, and not all sexagesimal; more uniform written sexagesimal writing system.
There is no indication in the symbols about how people counted on their fingers.