Your summary of what the book says is directly at odds with data that I've seen in books like Enlightenment Now.

https://towardsdatascience.com/has-global-violence-declined-... does a good job of summarizing the data. Every archeological site on record and every primitive society we have been able to study records a rate of death by human violence that exceeds by orders of magnitude what is seen in modern civilized societies. In fact they all exceed what happened in the USA+Europe from 1900-1960, and usually by a large multiple. Even with 2 world wars, Western civilization killed fewer per capita in violence than any previous period in history that we have data for.

In short, Hobbes was right. Every attempt that I have seen at taking a quantitative approach has backed that up.