In response to your main issue regarding socioeconomic mobility - why is this, in turn, a problem?
If you can demonstrate that reduced social mobility is an issue, and that this is increased by progressively greater income inequality (the latter of which I probably agree with on cursory inspection), then it seems income inequality is a net loss.
However, from a utilitarian perspective, I don't see a reason to care about social mobility if quality of life increases for everyone, not just the upper classes.
Keep in mind that I am working on two axioms:
1. Income inequality is a mathematical certainty in a functional society with more than n participants, where n is the maximum number of people who can be personally and honestly cooperative with each other, and
2. Income inequality refers to differences in an individual's absolute resources available for consumption, not their quality of life (exceptions granted for luxury consumption goods, not basic utilities that are available across all social classes).
As a very simplistic example that I'd enjoy having holes poked in, Apple has generated wondrous income inequality for many of its engineers and management. However, it has also leveraged economies of scale and competitive forces to manufacture a mass market product that is, statistically, enjoyed by even the poorest households in America and which affords its users with the most powerful innovation since the printing press in their pockets.
EDIT: This comment has had a rapid change in up and down votes. It's fine to disagree with me, but I don't think it's unreasonable to question whether or not a lack of social mobility is actually a bad thing, as a Socratic baseline.