It is popular in particular on the right to dismiss static inequality as relevant. The only thing that matters to them is equality of opportunity. Of course the world is not that simple. The two can not be completely separated. Children of the wealthy will have a major headstart on everybody else. Children of the poor will start with a disadvantage. Thus if you get rich in your 50s, that is raher irrelevant if you were poor when you raised your kids. They would have gotten off to a worse start in life. The distribution of your wealth or income through life is not irrelevant. Through the different phases of our life , the income requirements vary. People need more money as young parents and seniors than in other parts of life. Young parents because they have expenses for children and need a bigger house to fit them. Seniors because they will have more health care bills.
Thus for the average person I don’t think it is irrelevant how their wealth is distributed. Few would want to be very rich in their 40s and beggarly poor in retirement or their 30s.
Also the premise the inequality is really just about how much you respect those who got more than you, seems like a rather simplistic understanding of the problems of inequality. All the way back to 1776, Adam Smith understood that inequality and poverity is ultimately about shame, and not envy. Smith remarked that an Englishman was expected to wear leather shoes while a frenchman could walk barefoot without being ashamed.
The society we live in sets the standards of expectations. And that is indeed a zero sum game. By raising the overall wealth of a country, you also raise expectations. That is why people in a rich country like the UK could be going hungry while people in a poorer country don’t. The UK resident might have chosen to spend all their money on particular kinds of clothes or living conditions, which a family in a poorer country don’t have to, because expectations are lower.
Inequality also has major effects on happiness. While a country is poor, absolute wealth increase has the strongest effect on the happiness of a population. But as a country gets richer that effects tappers off and inequality starts dominating.
This is significant, because any society has to ask itself what is the purpose of anything? Why do we strive towards a wealthier country? Primarily because wealth enables us to provide better education, health care, happiness etc. Wealth alone has no purpose. It is only what we can do with it that matters. GDP is not a goal in itself. People overeating on junk food, getting obese and going more to the hospital will show up as higher GDP output. But that can’t be considered a good thing.
It is particularly relevant when discussing the US, as it is one of the wealthiest western countries and yet score below quite a number of metrics on the well being of its population against significantly poorer countries. Typically people in more statically equal societies have better outcomes on a range of factors.
But really we have only touched a handful of the issues with static inequality. Concentration of wealth has a tendency to also concentrate political power and undermine democracy. We get large corporations which are too big to fail and which puts the financial system at risk.
Not to mention the apparent propensity for asset bubbles to form in highly unequal societies.
I think any study of the subject will reveal that inequality has numerous negative consequences for society and envy towards the rich is probably at the bottom of the list.