What is the wisdom behind uttering the intention (niyyah) aloud in Hajj and Umrah, given that uttering it is an innovation (bid'ah)?

The intention resides in the heart. Uttering it verbally is an innovation (bid'ah) that was not established by the Prophet, peace and blessings be upon him, nor by his Companions. The Sunnah, therefore, is to form the intention in the heart, for Allah knows what is secret and even more hidden. Verbalizing the intention was not reported from the Prophet, peace and blessings be upon him, in any of his acts of worship, not even in Hajj or Umrah. It was not authentically reported that he ever said, "O Allah, I intend such and such." As for Dhuba'ah bint Az-Zubair's condition for Hajj, it was stated verbally because the commitment to Hajj is akin to a vow, and a vow is expressed verbally. The Prophet's saying, "Say: 'Umrah within Hajj,'" is a mention of the ritual during the Talbiyah and not a verbalization of the intention.