Due to: you might wonder if this is a verbal group, but it’s not—it’s one of those weird standalone phrases in English that has a very precise meaning and usage. (If you’re confused, just think about whether or not you could use the word due as a verb in other sentences.)
The beginnings of a new era of choice for Europe’s workers: this is a big nominalisation! The main noun is “era”. Everything on either side of that is adding detail to that noun.
For Europe’s workers: this is a qualifier within a qualifier (era of choice for Europe’s workers) both of which relate to the noun “era”.
On the basis of their skills or aptitude: as with “in the nineteenth century” this is a prepositional phrase (because it begins with “on”) so again to be consistent with other lessons we won’t highlight it. But I wish we could highlight it because the phrase “the basis of their skills or aptitude” is a big nominalisation!
(In fact the whole phrase “rewarding people on the basis of their skills or aptitude” is grammatically ambiguous because it’s a fragment. You could argue the whole thing should be one big nominalisation, but we won’t go into that here.)