My understanding of anarchism is the goal of eliminating government

The finer details will always change depending who you ask, but yes, it's generally either the elimination of government, or of all 'unjust hierarchies' (which includes state government).

As someone else mentioned, ideological anarchists tend to be socialists, and in this context 'anarchism' is assumed to be that socialist strain, but not everyone calling themselves an anarchist is also a socialist. It's a broad school of thought.

That won’t eliminate an economic system that originated organically.

Capitalism isn't organic. I can't think of a case where it has developed outside of a revolution (like the anti-monarchist revolutions) and/or imperial suppression. It requires the enclosure of the commons and development of private property security forces like a police, neither of those are an organic phenomenon.

If anything, I would assume anarchism is more organic, since it could be found in many hunter-gatherer gift economies. https://en.wikipedia.org/wiki/Primitive_communism#Example_societies

Now, I'm personally not convinced that this makes anarchism appropriate for our industrial/post-industrial societies, but it's not inorganic.

Anarchists are usually leftists... though not all of them are, some can be quite selfish

I think it's a beautiful ideology, but one that can't really stand up to imperialistic powers in the real world

No its the idea that authority/power is bad and we shouldn't have it.

Including cops, oligarchs, presidents, kings, popes, and sometimes even bed times.