Can a predicative clause be simplified to a gerund structure since both of them could be used as a predicative? Do "My idea is that the child should be sent to school" and "My idea is the child being sent to school" mean the same thing? Thanks a lot!
This is an excellent question—which, alas, has no very clear answer.
You are probably aware that different verbs ‘license’ different sorts of clausal complements: bare infinitive clauses, to infinitive clauses, for infinitive clauses, GEN- and ACC- gerund clauses, and that finite clauses.
The same thing is true of the sorts of predicative complements which may be attributed to nouns: each noun has its own preferences. And idea doesn't really like gerunds very much: gerunds are too “nouny”.
That is: with non-clause phrases idea is most comfortable with adjective phrases, including participles, which characterize the idea —good, original, ill-considered, depraved, intriguing ground-breaking. Noun phrases usually show up only when they express a characterization:
That idea’s a winner!
My idea is a novel approach to the problem.
This aversion to noun complements extends to clausal complements. When you speak of an idea as an action, a thing to be done, rather than a characterization, idea prefers infinitive and finite phrases, because these have a much “verbier” feel than gerunds.
My idea is to send the child to another school.
My idea is that we send the child to another school.
And if you want to cast the action in the passive you employ either your original construction, with a that phrase, or a for infinitive clause:
My idea is that the child be/should be sent to another school.
My idea is for the child to be sent to another school.
Noun clausal complements are like verb clausal complements: they must be learned word-by-word.