Your question is that of the Mogen Avraham !מגן אברהם סימן רסז ס"ק א
The Ketav Sofer responsa Ohr Hachaim siman 99 answers that the minchat chinuch holds like Rashi (רש"י, ברכות מח sv ad sheyochal kezayit) that the child has no rabbinic chiyuv. The chiyuv is chinuch on the father- this explains why in Berachot 20b the katan CANNOT be motzei a Gadol.
One can still ask further how the katan can be motzei a gadol with a chiyuv derabbanan (i.e as the gemara there suggests that the father ate less than "kedei Seviya".The ketav sofer answers this to (although I found his lashon rather cryptic) I understand/intuited his meaning as follows: whereas a person who is totally exempt from a Mitzva cannot be Motzei even a a kiyuv derabbanan, here at least the child has a "mitzva" under the rubric of chinuich which is a level up from pure "reshut" and therefore he can be motzei the gadol witha chiyuiv derabbanan.
This is the chidush of the ketav sofer which wraps up the sources nicely
The Maharam Shik שו"ת מהר"ם שיק או"ח ס' רסט (Rav Moshe Shik 1807-1879) also defends the Minchat Chinuch. Here is his "shakle vetariya" argumentation paraphrased as I understand it. Again his lashon is quite difficult (the acharonim wrote assuming their audience knew the suggayas of shas!)
Maharm Shik (M’’S) asks the following question on the Minchas chinuch
If the Minchas chinuch follows the reasoning of the Modechai in the name of R’Tuvia of Vienna then we are left to tend to the challenge of the Mogen Avraham (i.e your question!)
To answer this we say that the minor is not in the category of obligation and that is why in Brachot 20 he is unable to fulfil the obligation on behalf of the adult whose obligation is min hatorah. (this was the path of the Ketav Sofer above)
[here the Maharam Shik (M''S) challenged this answer]
If that is the case then similarly by the minor who becomes bar mitzvah during sefirah we should apply the rule that since a minor is not a 'bar chiyuva' then he cant fulfil his future biblical obligation from this point.
(only with a bar chiyua for example a blind man -can we apply the concept that he is able to fuilfil another's obligation since he will eventually become a fully fledged bar chiyuva deoraisa.)
[in other words answering the Mogen Avrahams question by invoking Rashi only helps to solve the gemara in Brachot 20b but leads us to question the case of the Katan shehigdil betoch sefirat haomer- and the minchat chinuch was using the Mordechai (in the name of Rav Tuvia of Vienna) to prove that the kattan dies continue counting with a blessing!]
The M’’S suggests the following answer: -
He suggests (a chidush no doubt) that the case in Brachot 20b (that a minor cannot be motzei a gadol who ate kedi seviya and has a biblical chiyuv to bentch) – goes according to the opinion in Sukkah 46a that we don’t make a blessing asher kidishanu on a rabbinic mitzvah. In other words there is no relationship between a rabbinic obligation and biblical one, and hence the child who is rabbinically obligated cannot fulfil the fathers biblical obligation on his behalf.
Note that according to this approach of M''S, we are no longer suggesting that the Minchat chinuch holds like Rashi that minor is not bar chiuyva.
However we pasken in halacha that we do make a bracha on a rabbinic obligation (Ner Chanukah is the proof see- sukkah 46a). The reason is that a Rabininc chiuy is rooted in a generic biblical one of “Lo Tasur”.
Based on this we can suggest that there is a relationship between the rabbinic obligation and biblical one, even though they are different levels of obligation, and therefore we can argue that the minor who is rabinincally obligated (not like Rashi who held he is not a bar chiyuva) can fulfil the Mitzva on behalf of another or on behalf of himself.
So this defends the Minchas Chinuch who suggests based on the reasoning of the Mordechai -that the counting performed by a minor prior to Bar Mitzva is a fulfilment of a rabbinic obligation and this (similar to making Kiddush during tosefet Shabbat) can create the Chiyuv deoraitta for when he is Bar Mitzva and so he should continue to count with a Bracha.