Bris Milah: Why Say ‘Aleinu After the Milah?

The recently published Sefer Kush’yos, by a student of the Maharam mi’Rutenberg, asks the question:

Why do we say ‘aleinu after the milah (i.e. not immediately after ashrei and u’va l’tziyon)?

Because in ‘aleinu we say: she'lo 'a'sa'nu k'go'yei ha'a'ra'tzos, “He has not made us like the nations of the earth.” In deference to the baby who is not yet circumcised, like a goy, we do not say ‘aleinu until the baby is circumcised, as well.
(Sefer Kush’yos edited by R. Ya’akov Yisrael Stal, Jerusalem, 2007, 5:9, p. 6f.)

This minhag is based on the custom that the milah takes place at the end of shacharis when ‘aleinu would normally be said. Once the milah is complete, the baby is then included in the tefillah of ‘aleinu, too.

The Aderes in his commentary on the Siddur, Tefilas David, feels that the minhag to have the bris before 'aleinu stems from the the custom to have the bris in shul for pirsum ha'mitzvah, publicizing the mitzvah. The reason for having it before 'aleinu was to be sure all the people were still in shul. (Jerusalem: Franco edition, 2004, p.51)