Is the sacrificial animal considered one of the reasons for exiting the state of ihram for the mutamatti' (one performing Hajj At-Tamattu') after shaving the head?

For Hajj, there are two stages of exiting the state of ihram (tahallul): The first tahallul is achieved by performing two out of three acts on the Day of Sacrifice (Yawm an-Nahr): stoning Jamrat al-Aqabah, shaving or shortening the hair, and performing Tawaf al-Ifadah. The second tahallul is achieved by performing all three acts. The preferred approach is to do them in order, and offering the sacrificial animal (hady) is not among the acts of tahallul.