The Simhat Kohen, the 9th Rosh Yeshiva of Beit El, writes in his sefer the issue of milah(siman 67, I believe) that we should pour the wine into the cup between the cut and the tearing of the Periah. Kabbalistically, this is where this step belongs in the various tikkunim that are taking place during the act of milah. He even brings a source from the Gemarra for this. However, he goes on, and says that we do not do this, again quoting a gemarra, because it would cause the child unnecessary suffering.
Essentially until the operation is complete and the child is bandaged, things hurt. There is actually Youtube footage of British Royal Marine, who did a show called Going Tribal. In one episode he underwent what we Jews call haphrada(where a metal instrument is used to separate the mucus membranes between the glans and the foreskin) and mahzikah(were we grasp the foreskin in such a way as to force the penis back into the body). He passed out from the pain in mid attempt, and they weren’t evening cutting anything. So make no mistake, circumcision hurts.
So according the Simhat Kohen when our holy sages were faced between arranging things in their proper order according to the metaphysics that are taking place, and causing a few more seconds of suffering to newborn, they chose to alleviate the child’s suffering and allow HaShem to worry about the flow of spiritual energies. There is a most profound lesson to be found in that.