I'm not sure this totally gets at the issue.
From a totally functional POV, of course families with fewer economic and social resources will tend to continue to remain in the same social class, while the children of those with more will tend to stay there.
And it's not even just economic, there are social reasons this happens - kids have some tendency to follow the way of life they see around them, and they tend to embrace a lot of the same values as their parents.
I'd go so far as to say, that so long as members of a social class are doing well enough to have economic stability, dignity in their way of living, and some political voice, many members of the group may not care all that much about upward social mobility.
The promise of social mobility is an interesting one, because it's not as clear as we might think who it is for. Most of the time advocates aren't proposing that we lose the jobs and social functions of the working classes. There will continue to be a working class, albeit receiving fair wages.
The possibility of social mobility for working class kids who might be interested in baking, or academia, or the arts, does benefit those individuals who have the talent and interest. But it's not clear to me that it has a lot kind of general benefit for the working class as a whole. (I won't say none - it means there will be some sympathy to and representation of views in certain sectors. As we saw in journalism in the old days.But it won't in itself make the wc positions unnecessary, so everyone will be middle class.)
I think the main benefit of social mobility is humanity - people are able to make the most of their talents, which benefits society. Though, it can mean the best minds are lost to lower prestige sectors - all the best minds become bankers, not farmers or mechanics.