In my experience, I've noticed that many people take for granted that if one needs a Melachah to be performed on Shabbath, one may have a minor, especially one under the age of 3, perform what is needed (eg., turning off a light left on in a bedroom Friday night).
It's apparently not widely known that because of a question raised by Tosafoth (Shabbath 122a), the Magen Avraham (325:22) infers that one may not benefit from Melachah performed by a child who has been asked to do it. The Peri Megadim takes this a bit further and understands that it is a really big problem to ask a child to perform Melachah.
But even the Peri Megadim, to the best of my knowledge, does not go so far as to ban the Hanaah (benefit) of Melachah performed by a child who does it on his own. This seems to parallel the Halachah of benefiting from Melachah performed by a gentile (and, indeed, this is, I believe, the generally accepted understanding - at least in my personal experience and limited research).
Would this not imply that the prohibition against having a child perform Melachah is of rabbinic origin? What about "Atah UVincha UVitecha" (Shemoth 20:9)?