What happens with trios, full trio, (full) semi-AFL, (full) AFL if we require closure under intersection?
Because it is a very natural set of operations: the full trio operations (morphism, inverse morphism, intersection with regular languages) exactly corresponds to closure under finite state transductions. 

We can formulate results for such nice families.
Requiring too many properties would exclude some families of languages. 
Who wants to miss the context-free languages?

Likewise the Boolean operations (intersection, union, complement) form a nice set, but we do not always require them.

The notion "intersection closed full AFL" is well studied.