At the end of the day, the difficulty with these comparisons is that social mobility measured in percentiles may be the same in a country where it is next to impossible to rise above your parents' social class as in a country where it is completely pointless to even want that.
In my country (Finland), for instance we don't force anyone to live in the streets. IMO, this creates huge amount of value from second order network effects - better childhoods, better social stability, people are more employable (if we only had jobs). I hope we can still afford this amount of social wellfare after a few decades.
In such a society, any meaningful definition of social mobility has to include more than just income percentiles.
So, we probably need prices and actual money for resource optimization, with human-sized goals for individuals to aspire to. But how to optimize human condition is partly a separate question and requires placing some value on the life experiences of individuals disrespective of their economic status.
To put it another way, if everyone made the same wage, an increase in salary of $1 would check the box for "social mobility" because it would move you into the top percentile.