I'm pretty sure that higher taxes in general do reduce inequality. Inequality is far smaller in Europe than in the US, smaller still in Scandinavia.
Also, there are European countries like Germany with an income distribution pretty close to the US.
Wealth inequality is more about the influence money has in both micro and macro-economic and social contexts than "I want what they have". Its no different than how other monopolizations work - the more of a supply you hold, the more distorted the market is around you. The same applies to money. Except money impacts all aspects of society, all markets - it gives you a greater voice in government, gives you influence over the lives of upwards of billions of people, etc. Just taking away that power from private individuals is often by itself a net positive for liberty.