Situation: "People living on a certain well-off island discover a way to farm more efficiently, which means they produce 10% more food, but that technology only works on their island. Is this good or bad?"
Inequality reducer: "This is bad, because that island's average income will go up while the rest of the world stays the same"
Situation: "We discover a free, simple cure for Alzheimer's"
Inequality reducer: "That's horrible, because Alzheimer's does the most damage to highly educated countries - those countries will now have way more mentally together, experienced 70-year-olds making them even richer, which will increase economic inequality with poorer countries where fewer people suffer from alzheimer's"
You have got to see this is a bit weird, right? It's a focus on minimizing local distortions sometimes can have a bad effect. In fact, imagine we really did have "equality" - the anti-inequality advocate would object to any change in that world, since it would lead to a temporary bit of inequality as information about the change propagated through it. Even simple things like the first invention of mechanized agriculture, or machines to make clothing, introduced tons of inequality at first, since there was only one copy of them at first! But we're better off now that our civilization doesn't have to devote 50% of everyone's life to growing food & making clothes by hand!
The other way to think about it is that that we just have to make sure everyone has at least a basic level of food, health, education, and freedom. That lets us focus on the main problems (people who don't have enough), and never could lead us into condemning gains in human quality of life.