Is the mitzvah on the taf or person who brings him. see tsfs here with rashash, chagigah 3, new minchas chinuch, Aderet ( I dont know who this is but saw him brought in a seifer). Does it depend on how old the definition of a taf is.
Thank you for making life a lot easier for me by providing the sources. I usually have to work harder by finding them myself! So all I have to do now is to summarize the opinions.
(1) As you noticed, the Gemara 34b states that the Taf are "Chayavim" but the Rashash points out that this must mean that the fathers are obligated to bring them, as the Gemara (Chagigah 3a) states that the Taf came in order to provide reward for those who brought them. So the Mitzvah is certainly on the person who brings them.
(2) This is also stated by the Minchas Chinuch Mitzvah 612:4 DH v'Taf, that you cited, who writes that the Taf were not commanded on this, because they are not old enough to be commanded. However I wish to point out a contradiction in the Mincahs Chinuch because in Mitzvah 17:14 (discussing the question whether a minor who possesses slaves who have not been circumcised may eat the Pesach sacrifice) the Minchas Chinuch writes that "in fact a minor is a Ben Mitzvah, only one cannot say the Torah warned him about Mitzvos because he does not yet possess the necessary intelligence, but in reality a minor is obliged in Mitzvos." The latter statement of the Minchas Chinuch is a very big Chidush and is quite a revolutionary concept.
(3) However there are other sources for the above concept. See Reb Chaim Brisker on the Rambam Hilchos Korban Pesach 5:7 Dh v'Asher who explains that a minor has a connection with the Dinim of the Pesach sacrifice. See also Shach YD 1:27 who writes that a minor is commanded on the Mitzvah of Shechitah.
(4) I will leave the above discussion for the moment as it is rather complicated. Now back to the Minchas Chinuch 612:4 who, as you saw, writes that one has to bring the Taf as soon as they are born. However the notes in the new Minchas Chinuch cites the Ramban on Chumash that the Taf the Torah refers to are not so very young but rather are children who are approaching the age when they can understand the Torah they will hear there. This is what the Torah (Devarim 31:12) states "In order that they should learn to fear Hash-m" which suggests that in the future they will acquire Yir'as Shamayim through hearing the Torah at Hakhel.
(5) On the basis of the above, it may now be possible to suggest an answer to the contradiction in the Minchas Chinuch that I mentioned above in (2). It could be that the reason the Minchas Chinuch 612:4 writes that the Taf are not old enough to be commanded is because the Minchas Chinuch is consistent with his opinion that Taf means 1-day old babies. In contrast the sort of minor that the Minchas Chinuch 17:14 refers to is slightly older and already has some understanding of the meaning of the Mitzvah. (In other words, for the Taf that the Ramban refers to, it may be that the Minchas Chinuch will hold that they have a Mitzvah).
(6) Aderet is an acronym for A.D.R.T. - Rabbi Avraham David Rabinovitz Teomim, who was one of the Gedolim about 100 years ago. He was appointed the Rabbi of Yerusahalyim (if I remember rightly he was the Rabbi of Ponevitch in Lithuania before that) but he passed away after only 3 years of being the Rav here. Please let me know what Sefer you saw the Aderet quoted in, so I can look it up.