QE and other aspects of Fed policy increased inequality pretty significantly. This is reinforced if you take into account all the other non-standard measures the Fed used to bail out the banks early on in the [2008] crisis.

Critics, often for good reason, are concerned that the Fed is wielding its vast powers in the interests of the banks and not in the interests of the people. After the financial crisis, Americans have perceived that the banks have been bailed out, but a significant proportion of the population is still in serious economic trouble.

Many of those on the right distrust the Fed and want to eliminate its power in the belief that the private economy, including the private banks, will be much more efficient, productive and even democratic if they are left to themselves: in other words, the criticism of the Fed really reflects a desire to cripple the government in the service of increasing the power and authority of the market.

Here's the interesting thing: the fact that QE and lowering interest rates almost to zero has worsened inequality, does not mean that raising interest rates will help reduce inequality.

Initially, QE contributed to a pretty significant increase in inequality. It raised asset prices, which are owned primarily by the wealthy, while having relatively small if any positive impacts on bank lending, employment, wages or economic growth, so ordinary people haven't had much help. By the third round of QE in 2012-2014, the effects had likely muted quite a bit. There were probably not big impacts on asset prices from QE and the positive effects on employment growth might have strengthened somewhat.

The right wants to destroy the power of the Fed to increase the power of finance; and the progressives want to reorient the Fed so that it will stop protecting the interests of finance and protect the interests of the broader population instead.

Raising interest rates might lead to some initial reductions in wealth by lowering asset prices, but it could also take a bite out of your paycheck and dampen your prospects of finding a job. It's a bit of damned if you do and damned if you don't.