Milah opens the eye to see Hashem

במדרש שמילה בשמיני שיראה פני מטרונה תחלה.

In the Midrash, that the milah (circumcision) is on the eighth day so that it should first "see the face of the matrona (the queen)."

The Sefas Emes cites the Midrash that milah waits until the eighth day so the child first passes through a Shabbos and "sees the face of the queen" — an allusion to the Shechinah.

משמע שיש אח"כ ראי' גדולה מזו כמ"ש יראה כל זכורך והוא באמת העדות שבנ"י בזכות המילה היו יכולין להיראות פני ה' ג"פ בשנה.

It implies that afterward there is a greater "seeing" than this, as it is written, "all your males shall be seen" (Shemos 23:17) — and this is truly the testimony that, in the merit of milah, Bnei Yisrael were able to be seen before Hashem three times a year.

The "seeing" at the bris is only a first stage; a higher revelation follows in the mitzvah of re'iyah (appearing at the Mikdash on the festivals), which the merit of milah makes possible.

ולכן יש פלוגתא בגמרא אי ערל חייב בראי' ע"ש.

Therefore there is a dispute in the Gemara whether one who is uncircumcised (areil) is obligated in the mitzvah of re'iyah — see there.

Because the festival "seeing" flows from the bris, the Gemara debates whether someone lacking milah can even be obligated in appearing before Hashem.

וקבלת שבת היא ראי' הקודמת כמ"ש הרמז שבעת ימים תחת אמו כו'.

And the receiving of Shabbos is the prior "seeing," as alluded to in "seven days it shall be under its mother" (Vayikra 22:27).

The first, preparatory "seeing" is the Shabbos the infant passes through before the bris, hinted at by the requirement that an animal remain seven days with its mother before being eligible for korban.

וכן בפרשה זו וירא אליו ה' ע"י המילה הוא עצמו ענין ראיית פנים בעזרה ועיין בסתרי תורה בזוהר בפרשה זו ותבין.

And likewise in this parsha, "And Hashem appeared to him" (Bereishis 18:1) — through the milah — this itself is the matter of "seeing the face" in the Azarah (the Temple courtyard); look in the Sisrei Torah of the Zohar on this parsha and you will understand.

Hashem's revelation to Avraham right after his bris is the same phenomenon as appearing before Hashem in the Bais Hamikdash — milah opens the eye to behold the divine Presence, as the Zohar elaborates.

ובמד' תנחומא פרשה זו בפסוק זרע יעבדנו יסופר לה' לדור מפרש שע"י התיקון שנעשה בהילד ע"י המילה שהיא עבודה בדמים כקרבן ע"ש עי"ז נזכר לפני ה' ונכנס בכלל ס"ר נשמתין דישראל:

And in the Midrash Tanchuma on this parsha, on the passuk "a seed shall serve Him; it shall be told of Hashem to the generation" (Tehillim 22:31), it explains that through the tikkun (rectification) accomplished in the child by means of the milah — which is an avodah in blood, like a korban — see there — through this he is remembered before Hashem and enters into the category of the six hundred thousand souls of Yisrael (shishim ribo nishmasin d'Yisrael).

Milah is a service "in blood," akin to a sacrificial offering; through this tikkun the child is "remembered" before Hashem and is brought into the collective body of the 600,000 root-souls of Klal Yisrael.

Summary: Milah on the eighth day, after the child has passed through a Shabbos, opens his eyes to "see the face of the Shechinah" — a first stage of the greater revelation of appearing before Hashem at the Mikdash; as a sacrificial avodah "in blood," milah effects a tikkun that has the child remembered before Hashem and joined into the 600,000 root-souls of Klal Yisrael.