Is this the reality at least in Europe? Especially with regards to people not leaving parents' houses, marrying much later, having less kids, and generally having less disposable income.

There always was some upward social mobility and that is usually called IIRC a safety valve. Very bright and fortunate can come from rags to riches, but honestly there are very very few. But from what I remember from my sociology book, the large majority of people stay the same or, go to +1 or -1 social class level (if there are about 7) compared to their parents.