Why wealth inequality becomes a problem...
Sometimes "wealth inequality" is a problem; and sometimes, it is just something that normally happens, even when everything is going about as well as it ever could.
Ancient Greece was an amazing laboratory of government, with dozens of small city-states that would periodically swing between monarchy, dictatorship, aristocracy, oligarchy, and democracy. Aristotle wrote about this in his Politics, which became a central text to the Founders of the United States.
Some city-states got into a destructive cycle between oligarchs and democrats. The oligarchs would arrange things to exploit the common man; the common man would eventually kill off the oligarchs, and take all their property. This was a double cycle of destruction, and resulted in impoverishment. The result was that the city-states became weak, and were eventually taken over and ruled by the monarchs of Macedon, who provided some much-needed political stability.
Thus, "wealth inequality" was a problem when the oligarchs "exploited" the common man. This "exploitation" was easy to see, or define, because it meant the impoverishment and worsening conditions of most of the population. This was typically achieved with concentration of agriculture, forcing the freehold farmer to become a tenant serf of absentee landlords; usury in the form of exploitive lending (often leading to the loss of property and serfdom); and exploitive taxation.
Short-lived democracies strove to cancel debts, reduce taxes and redistribute farmland. But, governments based on theft, plunder and abrogation of contracts typically did not last for long. Soon, people welcomed back the oligarchs, to kill off the democrats and establish private property rights again.
Today, "wealth inequality" might be a problem when it means a tendency for the common man's situation to worsen, while the rich get richer. What we would rather have is the Capitalist Ideal of everyone getting wealthy together. The ideal capitalist plutocrat built a giant company that provides a mountain of new goods and services at an attractive price, increasing the overall wealth of the society while employing tens of thousands of workers at high wages. Every economist will tell you that wealth is a function of production. The creator of this new productivity, the capitalist and business manager, typically does not enjoy very much of it. The average corporate profit margin, in US history, has been about 6% of sales. The other 94% effectively goes to workers, and the government.
For the most part, the US economy achieves the Capitalist Ideal, where all levels of society benefit. But, we should always be on the lookout for those situations where the common man's situation is worsening. Conservatives tend to have a blind spot here. Usually it is pretty easy to see: You just take Bernie Sanders' campaign platform.
Agricultural land ownership does not have the central role today that it often did in the past. Nearly all of the economy is non-agricultural. But, we do have an issue today with residential land ownership, enforced by zoning laws and other means. Housing costs are way too high in many locales, and the basic reason is that the common man (indirectly via property developers) can't build houses where the jobs are. There are other key sectors of the economy that look a lot like "exploitation." The most obvious is healthcare, which is way too expensive. Where free-market healthcare exists (hint: there's a price), it tends to be about 80% cheaper. One-fifth the price. Education is another one, now that college education, which seems practically useless, has been taken as a necessary for economic life. While "rentiers" referred in the past to absentee agricultural landowners, the healthcare and education sectors look like today's biggest "rentiers."
Today's bankruptcy laws, which permit a personal "Debt Jubilee" according to the Biblical seven years, have reduced the harm of exploitive usury. Nevertheless, many Americans find themselves in way too much debt, much of it non-dischargeable student debt, and also high-interest credit cards. In the 1960s, many States had usury laws that banned any consumer lending at rates over 10%. Attempts to make housing "affordable" have tended to encourage lending way too much money to people, compared to their income. It would be better to make housing cheap, by reducing those barriers that result in housing costs far higher than construction costs.
Taxation today does not follow the pattern of "oligarchical" exploitive taxation (for example in France before the Revolution), but rather "democratic" wealth confiscation. Just as was the case in Ancient Greece, this tends to depress economic productivity, and overall societal wealth. With the Capitalist Ideal of shared wealth-creation rendered difficult, things get worse for the common man. This is why the Founders, keeping Aristotle in mind, effectively banned Direct Taxation (income taxes) in the Constitution. Taxes should be indirect, and uniform, just as they were before the Sixteenth Amendment in 1913. Income taxes have been a battleground between "oligarchical" and "democratic" elements ever since, with general impoverishment the end result.
The "oligarchical" pattern of taxation and usury is more easily perceived in the many countries that have found themselves indebted to the IMF and World Bank, who then tend to impose high taxes to pay the debts. Unlike consumer debts, which can disappear in bankruptcy if things get out of hand, these sovereign debts are hard to discharge. They are ultimately paid by the common man.
Another key element of "exploitation" in recent decades has been globalization of trade, and unrestricted immigration. This has driven down wages for lower-income Americans, while driving down prices and employee costs for the higher incomes and capitalist class. A large swath of Americans – maybe, most of them – have not been enjoying the steady improvement in their condition that is promised by the Capitalist Ideal. In practice, this means lots of job opportunities, at higher wages. You could argue that it has resulted in better jobs and wages for people in China, or illegal immigrants from Latin America. But don't be surprised if you can't base government legitimacy on these claims.
The Capitalist Ideal should result in improving conditions for everyone, and tremendous wealth for a few. Nobody is very envious of the plutocrat's yacht, when they are able to buy a nice bass boat for the first time. A rising tide should lift all boats. This is not always happening today, so fix it.