I agree partly. Some inequality is certainly immoral, particularly when it's based on a person's race or gender. (And that definitely occurs in the US). But sometimes it comes down to other factors, like intelligence, work ethic, assertiveness, etc. if that's the cause of the inequality and I don't think it's desirable to eliminate it.

Why would those things change who gets paid more for the same work, or who has access to healthcare, or who is going to be hassled on the way to the voting booth, or who gets taken seriously when trying to buy a house, etc.

I never made any of those claims, nor do I endorse any of them. I will say that the concept of "same work" is very hard to define, and I don't think you can do so outside of very specific industries that are mostly low skill.