Avudraham (Birchos Nisuin) quotes Siddur R' Saadiah Gaon that there is (or was) not just a special haftarah for a chassan, but a special reading as well: Bereishis 24:1-7, which describes Avraham sending his servant to find a wife for Yitzchak. Rabbeinu Bechayei (ibid. 24:3) gives a reason for this:
ועל כן הוקבע בישראל מנהג לקרוא פרשה זו לחתן ביום חתונת להזכיר העם שיזהר בנשואיו, ושלא יקח אשה לשם יופי, שהרי הכתוב אומר (משלי ל״א:ל׳) שקר החן והבל היופי, ולא לשם ממון, כי הממון עשה יעשה לו כנפים כנשר יעוף השמים, ולא לשם שררת הקרובים ובני משפחות שיהיה נעזר בהם ויוכל להשתרר, לפי שהוא נכשל ונענש בכל אחת ואחת משלש כוונות אלו, אבל יצטרך שתהיה כוונתו לשם שמים ושידבק במשפחה הגונה...
This is the reason that it has become the custom of the Jewish people to read this portion (in the presence of the assembled guests) to a bridegroom on the day when he gets married* to remind all the people to be very circumspect in their choice of life-partners. The groom is reminded not to marry a woman because of her physical beauty [...] He should also not marry a woman because she is wealthy [...] The groom should also not marry a woman because she is a member of a powerful and influential family [...] If these are a groom’s considerations in the choice of his wife he will be punished and will likely fail on all three counts. The guiding considerations when choosing a wife have to be reverence for G-d, choosing a family which enjoys a good reputation.
* Or, per Avudraham, on the following Shabbos.
(Keser Shem Tov (R' Shem Tov Gaguine) cites all of these sources and then some, and mentions that nowadays there don't seem to be any communities that read that Torah section from a Sefer Torah, but in various Sephardic communities they read and translate it from a Chumash.)
So the special haftarah went together with a special reading from the Torah. The reason given by Rabbeinu Bechayei naturally doesn't apply to a bris milah, so there was no special Torah reading for it, and without a special Torah reading there wasn't a special haftarah either.