Learn English – What sentence part is “visitors” in “I’m not allowed visitors”

It might by easier to think of "keep abreast of" as a verbal phrase which is syntactically similar to (but not equivalent to) "monitor"

"It is not possible to monitor the normal tides of acquisition."

Ok, here's the best I can come up with, although it may not be a perfect match. (I'm a bit confused about your first example from characteristic 4, because English treats the first sentence as two independent clauses joined by a conjunction.)

It appears to me that English distinguishes three kinds of the "complément de phrase" without giving them a common name. The three kinds are adverbs, adverbial clauses, and adverbial phrases. The distinction between the last two is that the clause, while subordinate, has both a subject and predicate, whereas the adverbial phrase lacks at least one of these items.

Many prepositional phrases are also adverbial phrases (e.g. in the town), but I'm not sure whether all of them can be classified as such. For example, prepositional phrases that are used to indicate indirect objects or possession may not be considered adverbial. If such prepositional phrases are not considered to be adverbial, then, adverbial phrases and clauses along with adverbs seem to match the French category of "complément de phrase" pretty well.

With regard to characteristic 3, note that location adverbs and phrases can cometimes be replaced by words like "here" and "there," which function in a way that is similar to a pronoun.