The Yenuka, Rabbi Shlomo Yehuda Beeri, Shlita revealed that the blessing over fragrance during the circumcision is mainly for the baby. The godfather, who holds the baby on his lap during the ceremony, should give the fragrance (usually Myrtle) to the baby to smell. It supports his connection to holiness. Before the baby came to this world, his soul was in the Garden of Eden, and the scents of this world are unbearable for him. A good scent relaxes the newborn’s mind and soul with the understanding that the world around him is good.
For a similar reason, we smell fragrance immediately after Shabbat because Shabbat is on the level of thoughts (עולם המחשבה), and the soul נשמה is connected to the נשימה breathing. The scent connects the light of Shabbat to the soul.

Notes by Zion;
During circumcision, we use Myrtle branch הדס. It is natural, has a pleasant aroma, and represents the center column (the balance of Right, Left, and Center, Abraham, Isaac, and Jacob).
The baby’s environment should be clean and pleasant to serve as a soft landing from the upper world into a little and painful body. The bad smell of this world reflects the many impurities around.
Please do not take this advice to the extreme. No chemical should touch the baby’s skin unless prescribed by a health professional.
One of the amazing scents I experience is the scent of a head of a newborn. It has a unique scent that can’t be compared to others. It could be an impression of a smell carried with the baby’s soul from the Garden of Eden.