Why is highly organized/systematized violence preferable?

Why are you under the impression that you have a choice?

Those that are violent will always exist, and will always attempt to leverage their willingness for violence, and those that are able to - through luck or circumstance - gain leverage will be able to use that leverage to consolidate more until they have a hegemony on violence.

That's simply an observational fact of human history.

Governments are a way of occupying that opportunity space with some structure that _organizes_ that otherwise disorganized "cut people's faces up and flay them alive" sort of violence, and replaces it with "you get to grumble about the taxman every year" sort of violence. The averager person loses more freedom when there isn't a government around. The average violent psychopath gains more freedom (to become the government) when there isn't a government around.

That's why. Because I, and most people, prefer the paying taxes to getting flayed alive for insulting the duke. The government is a _binding_ of the monarchy and the warlord class to rules. If you look at the history of western democracy, it's extremely obvious.

I highly suspect that one of the reasons that Americans speak in this way.. that government as an idea is inherently some nonsensical or flawed concept - is compensation for their own sense of futility and inability to effect change on their own government.

It's hard for them to reconcile their self-image as "free thinking exemplars for the rest of the world" with the idea that they don't actually have control over their society. So they default to the idea that "all government is bad". If government by definition is bad, then obviously you can't accuse Americans and American culture of being particularly poor at creating a government that serves its people: it's just a fundamental structural problem, not a cultural problem.

To use internet slang: it's a cope.

The earnest answer is that it's more predictable than random violence, and its predictability makes it somewhat more preventable.

That said, there is a tendency for a system to drift away from this predictability as it's subjected to "review" by people who really, really want a particular outcome, regardless of the systemically-proscribed conclusion. "Bespoke" judgments for edge cases undermines the principle of predictability, which makes a return to "random" "coercion" desirable for some (as those who coerce in anarchy generally have less absolute power than a large system does).

But then, how do you show mercy (people are driven to do so) in a zero-tolerance environment?

This is the tension.

Organized violence is self regulating if it wants to sustain itself.