Even if, by magic, you manage to get all the world on exactly the same income per family/person at one point in time, that equality will only last a moment. Add any amount of time with some random changes in society, environment, sunspots, whatever, and some parts of the world will decrease, some increase in income. One island might be the new hot vacation spot, the other might be unreachable due to storms for a month, or have a volcanic eruption. People might get bored and vote for some crazy asshat leader in some country. Whatever happens, the income distribution will, with time, spread out from the initial condition of equality, basic statistic says that it'll roughly look like a gaussian distribution getting wider over time (It won't be strictly gaussian, and look less so after a while, since there are boundary conditions and secondary effects like development aid going to the poorer).

We are not uniform, the environment is not uniform, resource distribution is not uniform, and even if you tried to make it so, it won't stick. Any non-isotropic planet will produce inequality.

For the example of the US being unique or special, that might not be permanent. And the usual explanations might not always hold. But in any kind of world we can imagine, there will always be some luckier and unluckier parts, resulting in inequality.

Now you can try to improve the standard of living all over the world (and of course people do that). But as long as e.g. the USA is still interested in improving their own standard of living, there will be a game of catchup that might be unwinnable. The only solution I see would be to cause e.g. the US to cease improving its standard of living. However, I'd consider that immoral and inhuman. Everyone, even the top half of the distribution, have the right to strive for something better.