We frequently see statements in halacha like "the minor's obligation to do the mitzva is only rabbinically imposed (mid'rabanan), because it's only because of chinuch (=the obligation to educate one's kids)". This is attributed on this site to Mishna B'rura and to Minchas Chinuch, though I'm sure it predates them.
This is even sometimes used to allow, e.g., a minor who ate his fill of bread (and would thus be divinely-imposed-obligation obligated in bircas hamazon were he an adult) to say bircas hamazon for an adult who did not eat his fill (and is thus obligated mid'rabanan). Thus we see (seemingly) that the minor's obligation is considered to be in the mitzva at hand and not in some third-party mitzva like chinuch or something (even if it's imposed because of chinuch).
Am I right in my understanding that there's an obligation on the minor?
If so, how can the rabbis possibly impose an obligation on a minor? He's not considered to have intelligence! How can someone without intelligence be obligated in anything?
And if not, how can the minor exempt an adult in any mitzva?