I think inequality's really a red herring in this debate. As PPs have said, there will always be inequality (at least in a society where money exists and people have some freedom about how to spend it). However, it's much easier to be relaxed about the ridiculous amounts of money footballers get paid if you have enough that you aren't worrying about money every day, than if you're having to use a foodbank to feed your children. I think what matters more is making sure that even the poorest people in society have enough for a decent standard of living.
The UBI experiments are really interesting. I hope that can be made to work - which is mostly a question of whether the sums add up, I think. In one sense it's not so different from a social security system, but the cultural change we might get from making it be fine to choose not to work, or to work less than full time, could be really positive. For example, it's just silly that in some cases families would like to do more of the hands-on caring of their children or elderly relatives, but can't, because they can't make ends meet without working too many hours to make the caring possible.
There will always be difficulties - e.g. I think the argument for giving money not foodstamps is compelling, but there will always be people who, for a variety of sometimes complex reasons, can't manage with even what looks like a reasonable amount of money and still be able to afford food. Even UBI wouldn't make that go away.
I think we can and should be able to get to a point where almost everyone can get by reasonably comfortably on what they have, though. To do that we'll certainly have to raise taxes on the rich which will tend to reduce inequality - but I think we should see that as a consequence of what we do for a good aim, not as an aim in itself.
For lots of analysis on inequality it's interesting to read a pair of books in parallel: The Spirit Level and The Spirit Level Delusion.