We do not circumcise our male children because this was commanded to Avraham. We do this for the same reason that we do all the mitzvoth. Because it was commanded to the Jewish people at the giving of the Torah. It is in remembrance of the covenant of Avraham (Shabbat 137b, Yoma 85b, Chagigah 3a).
There are two general elements related to the Brit. One is the removal of the foreskin and the other is the shedding of ones blood (Mishnah Torah, Hilchot Milah 3:4, Shulchan Aruch, Yorah Deah 268:1,2,5). The foreskin aspect is associated with the covenant of Avraham, that we are his descendants, members of his household and the inheritors of his blessing (Shulchan Aruch, Even HaEzer 129:20). This is also why all converts are called "ben (or bat) Avraham", the children of Avraham.
The second aspect pertaining to the shedding of ones blood in service to HaShem is the idea of personal "mesirat nefesh" the dedication of ones life to serving G-d like we say in the Shema, "with all your heart (both your good nature and your bad inclination), and with all your soul (your life), and with all your might (your wealth). This is also the intention that we say in the Haggadah at Passover when it states twice, "By your blood you shall live. By your blood you shall live." Once relates to the blood of the Korban Pesach and the second to the blood from the circumcision. This is also the explanation found in the Yalkut Shimoni on Yechezkel 354.
Like it states there quoting from Shemot 12:44, 48, non-Jews and the uncircumcised could not eat from the Korban Pesach at the time of the first redemption.
In this context, the distinction made by the Ittur and the earlier poskim is understood. That even when a non-Jew who has been circumcised by non-Jews comes to convert, he must have the drop of blood taken. Because this is the declaration of his dedication to serving G-d.