I think also the period Dickens writes about (usually 20-30 yrs before his present time) is one in which childhood is being ‘invented’ that there is a phase between being born and adulthood. Dickens, as noted above, seems finely tuned to that moment when the world reveals itself to young eyes - the hypocrisies, the venality and the loss of innocence - it is a view borne of Romanticism but tempered by a belief in hard work, industry, self improvement, making one’s own luck.
As for school in itself, invariably it is a middle class luxury, I’m not certain but compulsory schooling isn’t introduced during Dickens’ lifetime. And in Nickleby Dickens makes a point of highlighting the brutality of schools after a national scandal in which essentially unlicensed schools visited such violence on their pupils that a number were killed.