Social mobility in and of itself isn't desireable. But the possibility for it, is. Otherwise you'd (1) live in a static society where those in power, stay in power, and those who're not, never will, even if they wanted to. And (2) you'd also live in a society which is not conducive to exploiting the talents and resources within it, as swathes of people are never able to move up the socioeconomic ladder.
When looking at an individual, you of course can't conclude that because someone didn't move up the socioeconomic ladder, that there's some kind of problem to be remedied. This could have many reasons, many of which entirely reasonable.
But looking at it from a societal perspective, on aggregate, it is a useful proxy. You'd expect that among different population groups in different countries, humans would be roughly similar in their tendency to be ambitious, safe for some cultural differences. If one in country you see that there's very little social mobility compared to another (taking into account cultural differences), that's a good proxy to determine that the earlier mentioned problems exist, (1) the power imbalance and (2) not using human resources effectively.
So if you find that in the US, social mobility is lower than in Nordic countries, despite a stronger culture of 'you can be anything you want' and the 'american dream', that can be a useful indicator that there's a power imbalance and a system not conducive to letting people be the best version of themselves.
How any one individual chooses to live their life is a completely separate manner. We would have to assume that roughly, there will be about as many people with a disposition to life as you, in countries around the world. That's why ceteris paribus it's interesting to look at social mobility on a national level.
It's a bit like BMI not being useful for individuals, but being useful for larger studies.