In a system of exchange, even when the rules of how exchanges take place is completely random, you necessarily end up with wealth inequality:
http://www.decisionsciencenews.com/2017/06/19/counterintuiti...
Now, in capitalism, where exchanges can be influenced more by those who have more wealth to begin with, it's pretty obvious that it will naturally lead to extreme concentration of wealth. So I don't believe this question "screams for some quantitative analysis" but there you have it anyway.
The essence of capitalism is indeed the protection of private property and the resulting dynamics will lead to the aforementioned results. I don't know how you cannot see it.
As for inequality throughout history, it is actually a pretty recent phenomenon. Considering humans have been around for 100k+ years, and that we were all pretty equal up until around 6k years ago, inequality is very much an exception in history rather than a norm. And you can make a strong argument that inequality only really took place once humans became sedentary and started to implement a practice of private property.
This is also how I think about human nature. The "nature" of hoarding and greedily taking everything and leaving your fellow humans with barely enough to scrape by is not something we have always done. In fact, we only became so successful in nature originally due to our innate capacity for co-operation and our social behaviour and language. The fact that our "nature" for over 90% of our existence was very different to the brutality of capitalism that we see today goes to show that there is no fundamental "nature" that explains inequality. It is clearly a function of the current system rather than an inevitability resulting from the nature of its participants.