Who would care about the scale of inequality in a world where the poor were fed and comfortable? I don't think that inequality is a base problem, although I have heard arguments that (due to things like money in politics for example) it can lead to things that actually are problems. But that doesn't mean inequality is itself a problem, as the side-effects could go away if the right efforts were made to control them. It wouldn't bother me if Jeff Bezos owned entire planets, as long as my own finances were secure and his planetary ownership didn't result in any undue limitations of my own rights on Earth.
Because for the foreseeable future there will be resources that are important to people and are limited. If inequality gets so severe that some people are unable to get the resources that are important to them then they tend to get upset.
One of the most obvious examples in America right now is a house in a desirable part of the country. Buying a house has always been a major investment; however, housing prices are now so high relative to income that many people will never be able to purchase a house in one of America's modern cities.
Let's say that Jeff Bezos gets twice as rich while I stay the same. Inequality increases, but I'm not unhappy. Now, let's say that Jeff Bezos loses half his money on the same day I lose half of mine. Inequality stays the same, but I'm unhappy. Therefore I conclude that inequality doesn't bother me, and its absence is no comfort. (I'm using money as a stand-in for goods and services, of course. If inflation doubled the number of dollars we had, I wouldn't be any happier.)
Jeff Bezos losing half his wealth in a stock market crash wouldn't help my poor neighbor any more than it would help me, so I would say the same argument applies.
I would say that being able to own a house somewhere that is compatible with your overall goals in life[1] is a feature of a functioning society/economic system.
Whether that makes it a "right" depends on your stance on things like positive versus negative rights. I'm arguing that a system that doesn't provide that to regular people is broken regardless of whether it is a right or not.
[1] By goals in life I do not mean things like "I want to own my own island" or "I want a penthouse in Manhattan". Rather I mean something along the lines of "I want to live somewhere that I can go hiking everyday" or "I want to live in a cosmopolitan city".
"modern city", maybe not, but there's a world of difference between, say, a shack in the woods and an apartment with aircon/heating/insulation (as appropriate) and easy access to groceries.
I don't know where the line is, but if you say the basic right is just "house", and that gets enforced, someone's going to try to draw that line as low as they can. Probably a few developers try to argue that if everyone gets a house then zoning laws need to be relaxed so they can afford it, and (hyperbole) we're back to windowless apartments with one room per family and one bathroom per floor.
Evicting Jeff Bezos from wherever he lives wouldn't help a shack-dweller unless you gave Bezos' old house to the woodsman. So, I would say that the woodsman doesn't care about inequality, instead any benefit he could get would come from an improvement in his own fortunes (irrespective of what happens to anyone else). If inequality was the problem, then bad things happening to someone would help everyone who was worse off than them, but if Bill Gates gets cancer, nobody is better off.
The issue with inequality comes when someone uses it to consume a sufficiently disproportionate amount of the resources that there is not enough for everyone else.
For example, if a group of billionaires bought up all of the property in New York City that would then mean that there wouldn't be any left for anyone else.
Assuming that Bezos lives in a fairly large house (and/or has several houses), evicting him could help quite a lot of people previously living in small shacks. I grew up in a NY suburb and you could probably fit two families (three in the larger ones) in any house in the neighborhood without anyone being particularly uncomfortable, and with no construction beyond the conversion of a half-bathroom to a full-bathroom. Given funding to more thoroughly renovate, you might be able to manage three. And that's just upper-middle-class suburbia, not billionaire mansions.
