I have yet to see someone actually explain why income inequality by itself is a bad thing.
I'm not talking about situations where there is corruption, like certain African-region dictators with gold plated limos while their people die on the streets from starvation, or more common, politicians being bought off by companies and individuals.
Someone explain this harm to me, because from where I'm standing in a first world country, it seems to be just so much complaining over sour grapes.
The answer to your question lies in your second sentence. When income inequality becomes so great that the rich are easily able to use their money to do evil things to the non-rich, then income inequality has become a bad thing. So no, inequality isn't bad in itself, but in a context in which unequal wealth permits the rich to do things they really wouldn't be able to get away with were they not rich, then something is wrong. I guess I'm assuming we agree that rich people doing evil things to poor people is a bad thing. Reasonable people might disagree about what exactly is evil and who exactly is rich, but it seems pretty safe to speak in generalities here.
In your thought experiment with Bill's future, the problem is if the rich control all of the robots and only use them to take care of rich peoples' needs. I think that would be very bad for the paupers, and bad for society overall. If the rich guy has 100 robots and the pauper has 10 (and 10 is enough to take care of actual needs), then there isn't really a problem, or if the rich use their army of robots to take care of the poor, then we're probably okay.
Anyway, that's my opinion. I think there are a lot of opinions on this topic that have merit. I'd recommend against assuming that the only people who care about inequality are whiners and communists.