There have been some statistical analyses made of prehistoric human Left/Right- handed tendancy (notably the study reported by the BBC twenty years ago
Left Handedness common in Stoneage) all of which appear to indicate that the frequency/commonality of this trait has remained unchanged for at least tens of thousands of years:
In general the general population today about 12% are left-handed, though populations vary considerably, between 3 and 30%.
Because handedness has a genetic component the researchers wondered why the proportion of left-handers should have remained so constant over 30,000 years - the age of the oldest cave studied.
They suggest that because left-handedness is relatively rare it provides certain advantages over those who are right-handed, such as in solo and group fighting.
The researchers say their findings add to the evidence that the evolutionary forces that cause right- and left-handedness are independent of culture.
I do question their overall research conclusion (and let's see who else detects a flaw in the same way I do) but if we accept the premise of "no change" in Handedness for tens of thousands of years, the situation long before that was intriguing.
I'm sure that because left:right Handedness is provably an undifferentiated 50/50 amongst all of our cousin apes, there have been conclusions derived which hint strongly that the origins of speech (pre-heralded by early neurological brain hemispheric specialisation) were the major drivers for this, rather than evolutionary advantage for combat. I can't find a citation to support this assertion, but instinctively it seems logical.
Especially when it is remembered that speech and motor movement are inextricably linked whilst people 'paint shapes in the air' with their hands & bodies as they talk (cf my past-mentioned observation that non-native speakers of a given unfamiliar language tend to maintain a rigid non-physiocommunicative stance that reminds one of a stroke victim).
Physical interaction creates narrative, within & outwith both the producer & the witness.
Whether living, making or taking life, this is inherently what it means to be Homo Sapiens. War/actions make thoughts|words; thoughts|words precede/accompany/substitute for war/actions.
And the capacity / specialisation for being left *or* right handed is an emergent token of an inner depictive/speculative specialisation, wherein being left or right handed is neither advantageously right nor wrong.