We are currently learning the laws of Shabbos, and have arrived at the segment related to what is done differently when there is a Bris on Shabbos!
The Talmud says clearly (it's actually a Mishnah).
תלמוד בבלי מסכת שבת דף קלג עמוד א
עושין כל צרכי מילה בשבת
All needs of the circumcision can be performed on Shabbos.
The reason this is important is because circumcision, mitzvah (commandment) that it may be, is elective surgery. And while the elective surgery is required, we do not violate or desecrate the Sabbath through having elective surgery on the Holy Day. Therefore, such a procedure should take place on any other day of the week.
Why is there an exception for Shabbos? Because the verse says that when a woman gives birth to a boy, on the 8th day he is to be circumcised. (Vayikra 12:3) [This means only if she gives birth through birth canal and the 8th day is absolutely Shabbos, meaning the child had to have been born after the stars came out Friday night, and before the sun set on Saturday.] This 8th day business is so serious, that when it is clear that the baby's 8th day is Shabbos, his required 'elective' surgery takes precedence over the Holy Day.
In the context of addressing what are "צרכי מילה" I mentioned two things that I heard from my rebbe, Rabbi Moshe Tendler.
1. Metzitzah, which is a tradition Rav Tendler supports for it to be done in a sterile way, using a tube, as a vestige to a requirement of the Talmud, but which clearly has no necessity today as the Talmud clearly mentioned it as an act which had medicinal purposes, whose merit is clearly not supported by today's medical understanding - in his view, performing Metzitzah on Shabbos is a violation of Shabbos! [See here where Rabbi Moshe Feinstein noted that, in general, if Metzitzah were not done (accidentally, for example) there would be no need to revisit the wound, and the bris is kosher]
2. Using a marker - I have noted here - Magic of a Marker and here - Getting the Percentages Right (and much more critically here) of the IMPORTANCE of using a marker when circumcising, for the baby's sake, to get the circumcision as precise and even as possible. There is no question, in my mind, that marking the foreskin is the most important thing I do in preparation for a bris, because once the foreskin is pulled forward to apply the shield, it is impossible to know exactly where the edge of the foreskin is without that mark. Particularly since we don't want to remove more skin from the ventral side, I maintain that any mohel who scoffs at marking the baby is being foolish and is doing a disservice to the baby. Rabbi Tendler is of the opinion that marking the baby is a necessity of milah, and that just as the circumcision (a Shabbos violation of cutting, causing a wound, etc.) pushes aside Shabbos, marking the baby (a Shabbos violation of writing) pushes aside Shabbos.
Two days later, someone in the synagogue told me that his son is a mohel in Israel, and that his son completely disagrees with these two points. Mostly, he believes that marking the baby is unnecessary EVER, because mohels in Israel don't do it! To which I said, most mohels in Israel don't wear gloves when they operate - does this make their method correct? (It happens to be that his son was trained wearing gloves, so he wears gloves.) By and large, I think I've addressed most of his son's and my differences in this post.
As for metzitzah on Shabbos, it boils down to how we understand necessity, how the Talmud viewed metzitzah (is it a requirement or medical recommendation based on medical knowledge of the time), as well as how we punctuate the Mishnah - most notably the continuation of the Mishnah, the part which I did not provide above.
Bottom line: As Jews we prioritize Shabbos. Absolutely. But Shabbos comes every week, and every week we have an opportunity to experience it better than last week. A bris is a ONE TIME opportunity, and you really only have one shot at getting it right. This is why the rules of Shabbos (for things that could not have been prepared in advance) are suspended for the mohel/caregiver when it comes to the conflict with Shabbos that a bris presents. No one questions that the rules of Shabbos are set aside when it comes to saving a life. Certain rules of Shabbos should also be pushed aside in order not to ruin a life through an imperfect circumcision.
So let us understand priorities, and not have our Hashakafic differences determine what is best for the baby. Let reality determine what is best for baby.