If you were to approach this question with intellectual honesty, you would identify pretty quickly that there are far better ways to try to answer it.
Case-control methods, natural experiments, surveys of criminals, and meta-analyses of the prior.
Literally any method other than "pick 600 year period and say 'vibes shifted generally across a continent and then homicide went down'"
Of course this question has been studied extensively for decades and the current conclusion is: completely inconclusive!
There's some evidence it increases violent crime, some that it decreases it, most evidence doesn't clearly show any effect at all.
So whatever effect it may have, it almost certainly isn't very strong, or is countervailed by opposing effects.
I think that if we're proposing the State, which we know to be fallible in so many cases, should make irreversible decisions like "executing suspected bad guys" more frequently, then we should have extremely strong evidence that it would actually achieve the desired result.
> It's also at least quasi-testable; someone could fund a study on examining alleles associated with aggression in historical remains.
Good luck establishing how "alleles associated with aggression" contributes to violence. I'm pretty sure most of the people who adopt your position would argue that their "aggressiveness" is a virtue in whatever competitive landscape they choose to occupy.