Your question is very interesting. The Gemara in Kesuvos 11a states that Rav Huna said that a minor convert may be immersed in the Mikveh with the Da'as (knowledge) of Beis Din. The Gemara says that the reason is because it is a Zechus (merit) for the child to accept Judaism, so he can be converted even though he (because he is a minor) does not possess Da'as himself.

However, RASHI there DH AL DA'AS writes that the Gemara is discussing a child who does not possess a father but his mother brings him to Beis Din to convert him. TOSFOS KESUVOS 44a DH HA'GIYORES writes that if the mother did not bring the child, there is no reason why the Beis Din should convert him because Beis Din does not convert someone unless he demands it.

An interesting explanation for this is given by SUKAS DAVID (by Rabbi Kwiat of Mir Yeshiva in Brooklyn) KESUVOS 11a #6 DH UBE'DA'AS. Even though a minor does not possess Da'as, he does nevertheless possess "Rotzon" (desire). Even though it is a Zechus to convert, nevertheless it is only a Zechus if one wants this. One cannot force someone to accept a merit against his will. Even though it is clearly the greatest thing in the world to be Jewish, nevertheless there are also many restrictions and prohibitions and since the baby was born non-Jewish, he has absolutely no obligation to accept these, so we may be presenting him with a difficult dilemma later on when he grows up. Without accepting all the Mitzvos, conversion is invalid, so there is no middle option of accepting the easier parts of Judaism without also taking on the harder parts.

In fact when an adult candidate for conversion comes to Beis Din and asks to be converted, we try and dissuade him (see YEVAMOS 47a) and point out that he might be making his life a lot more difficult by this decision. Indeed the Gemara there (47b) states that Gerim are as difficult for Yisrael as leprosy (see different reasons for this in Tosfos DH KASHIM).

Therefore, the baby found in the city where most are non-Jews is left to be collected by a family from the majority population because we do not know if he really would want to be Jewish.

KESIVAH U'CHASIMAH TOVAH

D. Bloom

I found that your question is asked by SHITA MEKUBETZES KESUBOS top 15b in the name of SHITA YESHANA. He asks why Beis Din does not immerse the baby in the Mikvah and convert him? He gives 3 possible answers:

1. Beis Din only accepts child converts if the mother brings him (this is similar to the answer I gave last time based on Rashi), or if the child has the intelligence himself to request conversion. He adds that this Din only applies in Eretz Yisrael in the Temple times, but the new edition of SHITA MEKUBETZES note 148 writes that this is the addition of the censor.

2. The Gemara here is discussing a time or place where Beis Din is limited by local legislation concerning the reception of Gerim.

3. If Beis Din converts him he can still refuse, when he grows up, to accept Yiddishkeit (see Kesubos 11a) but people might still mistakenly think later on that he is a Yisrael.

However note 146 to Shita Mekubetzes cites RASHBA (quoted later by Shita Mekubetzes on p. 159 of the old edition DH VE'KASAV OD HA'RASHBA) who disagrees with SHITA YESHANA and maintains that Beis Din may accept a child even if his mother does not bring him. He writes therefore that even though a Jew may take the baby into his home, nevertheless one can say that the Gemara here is teaching the Din if no-one wanted to do this.

YEYASHER KOACH

D. Bloom