What does this Medrash teach us?
1. Only Hashem can allow us to serve Him, and He does so by stating the manner and granting us the means of doing so. Reb Moshe said that we learn two things: that we cannot invent novel means of serving Hashem. Hashem can only be served in a manner that He expressly sanctions. Anything else is "Sh'chutei Chutz," as if we brought a sacrifice outside the Beis Hamikdash, which is a cardinal sin. The second thing is that we can only approach Hashem after we have been imbued with a special kedusha, and it is only only after going through Shabbos that is one changed by a kedusha that makes avodas Hashem possible.
2. Bris Mila is a form of Korban. That Bris Mila shares certain characteristics of a sacrifice. Indeed, the Abarbanel says that the bris is a type of Korban. The reason it is done lechatchila on the eighth day and not later is because every korban needs to be without blemish, physically perfect. But this korban has less to do with the physical as it does with the spiritual. Thus, mila should be done on a neshama that is spiritually perfect. This is best done as soon as possible after birth, before the child has a chance to do what people are wont to do. Seize the moment when the neshama is still perfect.
So, why it that a bris milah can only take place after experiencing the kedusha of Shabbos? On the contrary! True, the experience of Shabbos invests us with kedusha, but shouldn't the departure of Shabbos result in Tumah? We have a neshama ye'seira on Shabbos. When it leaves us, shouldn't it bring Tumah in its wake? (also discussed in the Shem MiShmuel.) Life =Kedusha; Life ends, Tuma enters. Shabbos=Kedusha; Shabbos ends...... what should happen?
The answer is that when life departs, it leaves nothing behind. If anything, the object that has lost its life is worse than if it had never lived. That is not the case with kedusha. When kedusha leaves, some effect remains.
3(a)2. The greater the Tzadik, the greater the residual effect of his Kedusha. There has been a recurring assertion that the bodies of tzadikim gemurim do not become tamei. See, for example, Rabbeinu Chaim Kohen brought in Tosfos Kesuvos 103b DH Oso, and the Medrash brought in Tosfos Bava Metzia 114b, middle of the page, though Tosfos disagrees, and the Ramban in Chukas about Missas Neshika; The issue was exhaustively covered and conclusively laid to rest by Rabbi Marcus Spielman in his Tziyun L'Nefesh Tzvi, Brooklyn, 1976, in which he brings hundreds of mekoros on the topic, and more importantly the sefer has haskamos from Reb Moshe, Reb Yaakov, Rav Rudderman, Rav Hutmer, and Reb Shlomo Zalman Auerbach in which they all unequivocally state that le'halacha, we are not someich at all on the shittas hamatirim. (See a very nice review of the sources here.) But the point is that there seems to be some concept that there is less tuma on the bodies of Tzadikim. Why would this be so? On the contrary. According to the Akeida and the Zohar, there should be more! The answer is that Tzadikim convert their bodies into holy things, and even after their death, their bodies retain kedusha. The Malbim cited above says something very similar, as does the Shmaitsa in the Hakdama.
3(b). Each and every Shabbos is an opportunity to incorporate and concretize the spiritual growth of the previous week. When I was in Yeshiva, my Mashgiach and Rebbi, Reb Dovid Kronglas, knew that I hadn't gone to the Mikva before Rosh Hashanna and he knew why: I am of Lithuanian derivation, and going to the Mikva was not something men did. So he came over to me and said, "................., it is kedai to go to the mikva, it is brought in Shulchan Aruch, and every baal teshuva is required to go to the mikva, just as a geir must go to the mikva." I'm not sure it was he who added that "If tvila can make a goy into a Jew, imagine what it can do for a Jew!" I, being who I am, immediately decided that the example of a ger is irrelevant, because while going once can have an enormous effect, there is no difference between going once and going a million times. It's like annealing clay: once it's been in the kiln, it's not going to get any harder if you put it into the kiln another time. Or it's like hechsher for tuma. Once it was touched by water, it's muchshar forever.
Of course, I was wrong, and it's certainly a minhag tov to go to the mikva, at least once every year or two. But I'm not sure about the effect of Shabbos. We see from the Medrash that experiencing Shabbos is an enormously powerful spiritual event that forever changes whatever it touches. It makes a person fit to serve Hashem. It even makes a non-sentient animal fit to be offered as a sacrifice. But we don't see from the Medrash that the second Shabbos has any effect at all.
But according to what Reb Moshe said at the bris, it changes the whole meaning of the Medrash. If the idea of Pnei Matronisa applies to the Milu'im, then it must be that Shabbos enables growth in Kedusha not only for a newborn, but even for old wrung out shmattehs. Each and every Shabbos is an opportunity to incorporate and concretize the spiritual growth you worked for during that week.
4. You can make a Shalom Zachar on Yomtov instead of Shabbos. The Chasam Sofer (Toras Moshe Parshas Emor DH Mimacharas, the third one with that DH) says that if there's a Yomtov after the baby is born that comes before Shabbos, then the Yomtov does the same thing that Shabbos normally does. Theoretically, then, (according to the Taz in 265 brought in the beginning of this piece,) you ought to have the Shalom Zachar on the Yomtov and not wait till Shabbos. But it's best not to mix people up, unless you live in a community of Talmidei Chachamim who would enjoy the azus panim more than worry about the minhag.
5. It's a good thing Shalom Zachars are not by invitation only. This last piece, which speaks of the Shalom Zachar, is interesting, but best left in Yiddish, because it might lead to some very lonely Friday nights.