Communism is an organized form of government

Communism is the complete opposite of organised government.

Communism is "a theory or system of social organization in which all property is owned by the community and each person contributes and receives according to their ability and needs." (Oxford Dictionary.) There is no need for organised government in communism, let alone a coercive one. This is why there is such a close relationship between European anarchism and communism: so called "anarcho-communism".

The whole point of communism is that it is a community system, like in communes, just as you describe here:

a free trading market, perhaps without land ownership. ... people are more likely to share willingly, because they can all see how it benefits them ... People will work and hunt because they can see a direct benefit

Exactly. What you describe is the Utopian vision of a communist society, and we know that it works well in small groups. But it really isn't clear that it scales up to eight billion hairless apes, or even a few million in a single city, let alone large nations.

But even if it did scale, we still have the problem of how we're supposed to get there starting from the deeply unequal capitalist society we live in, where there is a parasitical elite class who owns the means of production, and has the money and power to protect their position at the top of the pyramid. Power never steps down without a fight.

Marx's solution to that was a transition period of socialism, where a strong government would defeat the capitalists and their hangers-on, institute government ownership of the means of production, and then, in the fullness of time, wither away leaving a pure communist utopia.

I think Marx was sharp as a knife at recognising the problems and flaws of 19th century capitalist society, but pretty naive about human nature. Marxists, in my opinion, are every bit as woolly-brained and unrealistic as American libertarians.

Short of a time machine to go back to before the invention of agriculture, it is hard to see how to reach that supposed Utopian communist state from the world we live it today. Marx thought it would be the workers of Germany and England that would start the revolution. But the capitalist class didn't just stand back and do nothing: by using a combination of carrot (pensions, more liberal and progressive laws, allowing unions to negotiate better pay and conditions, etc) and stick (crushing radical revolutionaries, by force if necessary) plus propaganda, they managed to avoid socialist revolutions in the west. Completely against Marx's theories, the only successful socialist revolutions have been in agrarian societies like Russia, China, Vietnam and Cuba.

And so today most people in the industrialised west live in some sort of welfare state, a mixed socialist/capitalist economy with safety nets, giving us the best of both systems. That allows the elites to keep their ill-gotten privileges, while still removing the need for radical revolution. This is why most "radical leftists" today are only looking to reform the system, not to revolutionise it.