Speculative phase:
When talking about base 60, the first thing that occurs to me is the notion of time: 60 seconds constitute a minute, 60 minutes constitute an hour, etc. However, it is unlikely that the Babylonian people had the technology to use such a precise and complicated time-telling system.
I then suspect that they used base 60 in combination with base 10 to reduce the amount of writing needed on the clay tablet, as it was difficult to store clay tablets of larger sizes.
In other times, sexagesimal is used not only in the time-telling system as mentioned above, but also in a variety of practices. For example, ancient Chinese started to use the sexagenary cycle to represent dates in cycles of sixty terms.
Research phase:
There have been numerous studies on why the Babylonian used sexagesimal counting system.
The Greek mathematician Theon of Alexandria (335-405) proposed that 60 was the smallest number divisible by 1, 2, 3, 4 and 5 so the number of divisors was maximized. While some people do not agree with this reasoning by saying that a base 12 system would be more efficient in terms of the number of divisors. I personally like Theon's theory because the number 5 is an important component of both the base 10 and base 60 systems. Moreover, humans have five fingers on each hand, making counting in five's a very practical methodology.
Otto Neugebauer (1899-1990) suggested that the adoption of sexagesimal by the Babylonian to divide weights and measures into thirds. Again, this theory does not explain why the base 12 system was not chosen instead.
Reference: https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/
You present some good arguments for the ways in which 5, 10, 12, and 60 are all key players in the Old Bablyonian sexigesimal system! There are also 12 knuckles on our four fingers.
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