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Are capitalism and socialism actually polar opposites - apples and oranges, black and white?

Yes.
Private control vs government control.

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What is the allure of capitalism?

Wealth, power, and the accumulation thereof.

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socialism can be an economic or a governing system.

Only inasmuch as capitalism can also be an economic or governing system.

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it is easy to make the case that practically all cultures function with a mix

Yeah it's pretty easy because all cultures function with a mix. It's human nature, it's tribal. Things fall apart when they become unbalanced. Our current system of economic governance has become too heavily loaded on the capitalist side. it's unbalanced and it's falling apart.
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"Be yourself; everyone else is already taken."— Oscar Wilde