My concern is less with income inequality per se and more with what the nation chooses to buy with its tax dollars, and what it chooses not to buy with its tax dollars.
After taking into account tax transfers, and comparing apples to apples across the board, we have a bigger upper class than any rich country (other than Luxembourg) and one of the smallest lower class percentages. We are in the middle-to-lower portion w/r/t size of the middle class. Those that are lower than us on that metric are not countries anybody aspires to imitate (Italy, Spain). Those that are significantly higher are the countries whose policies are seen as worthy of emulation (the Nordics).
The major difference between being middle class in America and middle class in Europe is about how much value you get from the state. In Europe, broadly speaking, you get high quality health care and education, and a secure retirement. In America, you get a really really big military. The American middle class is getting killed by the obscene cost of health care (whether directly through premiums or indirectly in cost paid by their owned businesses), by the cost of higher education, and for retirement, well, you're on your own to save as much as you can.
These are well-known problems. Nobody with any sense argues that the American health care, education, or retirement systems are just fine and in no need of reform, additional funding, or both. We have the largest upper class in the world, with the most money, and yet the argument is these poor rich people just need (yet another) tax cut and that trickle will come down and fix all these problems. Well, it won't. It hasn't for the past 30 years, and it is not going to start now. These problems are not amenable to market solutions, they need to be fixed by government policy, and that takes money. The American upper class is big, it has the money, and the rest of the country needs help a lot more than they do.