Increasing tax rates, even if it mostly on the rich, could theoretically lead to lower inequality levels but this decrease in inequality would be at the expense of sending the rich abroad and making them renounce their citizenship rather than making the lower classes richer. For example, Eduardo Saverin, the Facebook co-founder, who held double citizenship Brazilian-American, chose to renounce US citizenship and go live in Singapore. Brazil does not tax citizens working and living abroad and Singapore only tax residents for income made in Singapore. Therefore, if Eduardo has all his US$ 9.4 billion (Forbes) invested in a no-income-tax jurisdiction, like Monaco, Dubai, or the Bahamas, he possibly is paying 0% income taxation, legally. Just in the first year that Eduardo renounced his citizenship, he saved more than US$ 700 million in taxes (Sullivan, 2019). Additionally, to make tax havens like Singapore, Dubai, and the Bahamas, to adopt more stringent tax regulations is virtually impossible. Increasing tax rates, even if it is mostly on the rich, is unlikely to increase Americans' wealth.
What should be done then?
As I said earlier, the biggest problems in America are its culture and its political system. The biggest problem of inequality happens when the wealthy has more political power than the rest of the population because this can lead to a loss of freedom. That is, you can become pretty much a slave of the rich who has the political power. This is clearly a huge problem in the US. Political candidates who spend more money in their campaigns tend to be elected and the people who finance them are rich individuals/corporations looking for lobbying interests (e.g. Graetz and Shapiro, 2005). Moreover, mostly wealthy people run for the main public offices in America and the US elections are the most expensive in the world by far. According to the University of California - Berkeley Professor Gabriel Zucman, "in 1980 the average income of the bottom 50% in the USA was 16,000 dollars. In 2018, the real average annual income for this bottom 50%, i.e. after adjusting for inflation, is still 16,000 dollars. [...] So, although the US continued to grow all this growth has been taken by the top half. The bottom half of the American population has been shut off from economic growth in the last 30+ years." The biggest problem of inequality is on lack of freedom and it is a consequence of weak institutions where the rich have much more political power than the poor, although everyone has one vote (For more on that see "Inequality and Freedom").
If income inequality is to cause this poor American "serfdom," how can one explain Switzerland, for example? Switzerland not only has a larger number of billionaires per capita than the US, but they historically offer a de facto no income taxation program for the super richer, know as the Swiss lump-sum taxation. For example, Jorge Paulo Lemann, Brazilian richest man and one of the wealthiest people in the world (as of 2020 - Forbes), moved to Switzerland partly to virtually remove all his income taxation duties. Although Switzerland has a higher concentration of wealth than the US in several measures (Davies et al, 2007), the rich do not influence politics to the same extent that American wealthiest do. Despite the large concentration of wealth, income inequality in Switzerland has been flat, if not decreasing (Graph 1). In 2015, Swiss citizens were asked in a public referendum if they should tax the rich more and 71% of the population said "no." The US approach to decrease income inequality should be to make its political institutions stronger, i.e. more like Switzerland. Rather than increasing tax rates, what should be done are political reforms. Here is some example of potential improvements to the US political institutions:
Graph 1. Income Inequality US vs. Switzerland (1974 - 2017)
Source: The World Bank
Do I think enough changes like those will be made in our lifetime? No, because elections cannot fundamentally change a country. Major, rapid changes only happen during very rare periods and it is usually initiated by international shocks and not domestic decisions (e.g. Gunitsky, 2014). Elections are much less important than most people think. In the foreseeable future, US citizens will continue to support tax increases, especially on the rich, and they may succeed. However, this raise in tax duties will not decrease inequality by lifting the middle class as they hope. There is virtually no evidence of a country in the last 5000 years which solved the inequality by lifting the poor, historians say. All major income equalization came as a way of mass warfare, complete revolutions, state collapse, and catastrophic plagues (Scheidel, 2017). How many countries do you know that 50 years ago were highly unequal and today are highly equal? Instead, an increase in taxes would spike further US citizenship renunciation numbers. Last but not least, even if a tax on the rich could increase middle-class wealth, this would not make US citizens' lives better because the correlation between money and happiness is very blurry. More than changing their political institutions, Americans should change their culture, which is also a multi-generational change. If inequality really bothers you, the data says that your best move is to leave the country.