The Jewish rite is precisely specified. It takes place on the eighth day of life, counted inclusively, and it proceeds even when the eighth day falls on the Sabbath or the Day of Atonement, which very few obligations do. It is postponed only for the health of the child, and postponement for that reason is required rather than merely permitted.
The operator is a mohel, trained in both the procedure and the law governing it; many are also qualified physicians, and in some communities a doctor cuts while the mohel recites. The child is held by a sandak, an honoured role frequently given to a grandfather, and is placed briefly on a chair set aside for the prophet Elijah. Blessings are said by the father and by those present, the child is named, and the meal that follows is itself treated as a religious obligation rather than a social addition.
The theological content is covenantal rather than purificatory. The sign is placed in the body of a child far too young to consent, and that is the claim being made rather than an awkwardness to be explained away: membership of the people precedes individual decision and does not depend on it. This is exactly the feature modern objection fastens on, and the tradition has never pretended otherwise.
One element is contested inside Judaism itself. Oral suction of the wound is still practised by a minority of Haredi communities, and public health authorities in New York and in Israel have linked it to transmission of herpes simplex to infants, including deaths. Most Jewish legal authorities permit or require suction by pipette instead, holding that the requirement is for suction and not for the mouth. The dispute between the two positions is real and has not been resolved.