Inequality is irrelevant - reducing absolute levels of poverty is what matters.
Nope. Economic and social studies show time after time (try Joseph Stieglitz "The price of inequality" for one summary) that people feel worse when they measure themselves against those whom they consider their peers and come out worse.
That is why keeping up with the Jonses is a really interesting phenomenon. We can talk about absolute poverty and we definitely should be doing something to address that on a global scale.
But we should also be addressing the more important for people in the UK notion of relative poverty too. One of the way some countries more or less successfully address that is through income (and other) taxes. So that those who earn way above the national average pay more than those who earn below. Those who earn below are given grants and subsidies and benefits to bring them up closer to a level that we would find acceptable in our country.
And this has a lot to do with privilege. Who earns more? people who were born into wealthy families, had private education, went to good universities (sometimes on grades far worse than those achieved by children from less affluent backgrounds) and so on and so on. The fact that people tend to marry into similar class/backgrounds consolidates that.
Girls, as a group, used to do worse at school than boys in terms of exam success and entry to university etc. This was because boys were privileged and education more tailored to their strengths and needs. After a few decades of redressing the balance we see that girls do very well. Working class boys as a group, it seems, are doing worse at school than other socio-economic groups and it probably means that we need to tweak the balance. Or find some way of bringing them up. It need not be at the expense of other groups having to do worse (ie. It is not a zero-sum game)
The same could be said about lots of things: recruiting more BAME to the police; getting women or POC in higher positions in companies; blind recruitment; making sure that accessibility is foremost in our minds when planning spaces where people congregate for work and pleasure.
It doesn't take away from anyone but it does enrich society.