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Brit milah is one of the central mitzvot of Judaism, taking place when the baby is eight days old, if he is healthy, or as soon as possible thereafter if not. A brit is delayed whenever there is the slightest risk to the baby's health (Jaundice is the commonest reason for postponement).
The brit itself requires cutting the foreskin of the baby boy's penis, symbolising commitment to the eternal covenant between God and the Jewish people. Although the baby certainly experiences some discomfort, it is pretty clear to a parent that a moderate attack of burps is more painful to a baby than a brit milah.
Jews have fulfilled brit milah for over 3,000 years sometimes in the face of persecution. The Talmud says that all of the mitzvot observed by the Jews under threat of death are still widely observed today. It is a high and holy mission to be a Jew, but not one that we choose - the baby does not elect to have a brit. Our deepest commitments are already there before the ones that we ourselves choose.