I've noticed that libs tend to focus on economic metrics like "intergenerational economic mobility" instead of class when they discuss economic inequality. Basically instead of getting rid of classes altogether they want ''increased mobility between classes,'' or, more accurately, since they don't really focus on the relationship to the means of production at all, and instead they just focus on "income brackets" regardless of whether you own capital/land/means of production or not, they actually want ''increased mobility between income brackets.''
So they want it to be easier for a high earner to become low again, and a low earner to become high again. They view ''increased volatility'' in an individual's earnings over one lifetime as an ''indicator of a healthy society.'' because when so-called "economic mobility" is too low, class status becomes set in stone from birth to death, like slavery or serfdom, and it is harder to sell the prole on the "rags to riches" story that is used to justify capitalism as "free" and "full of opportunities for those who work hard." But how is this increased volatility of rich becoming poor and poor becoming rich an indicator of health? Doesn't that make society way more unpredictable? You wouldn't want that kind of volatility in the stock market, so why would you want it for "economic mobility?" Even from a capitalist perspective, wouldn't it make more sense to want either consistent ''growth of wealth'' across ''all income brackets'', i.e. poverty reduction, i.e. "rising tide lifts all boats"? This is what China aims for, for example.
I remember first being made to understand this issue kind of from reading an Existential Comics tweet that social mobility -- despite the vaunted place it gets in neoliberal rhetoric -- is an absurd thing to be worried about. It's plainly more important how conditions are at the bottom than the rate at which who specifically is at the bottom changes.
But to answer your question, the real reason is that normal liberals have much more in common with fascists than they want to admit and they fundamentally want a "meritocracy" where the skilled are saved and the inept are condemned, without ever asking why anyone needs to be condemned in the first place.
And capitalists of course prefer this model because it obfuscates things like their use of poverty as a tool of worker discipline (see "reserve army of labor") behind fables of hard work and hope.
It's Protestantism, if anyone is wondering. Their completely unexamined belief that some people will succeed because of hard work and some will fail because they're bad and lazy is just Calvinism. Purely religious belief, but they've secularized their Protestantism to the point where they're completely unaware of it.