Who was the most important in the destruction of Veii and what did he do?
Let me give you a little historical context first. The best account of Rome's fight with Veii comes from book 5 of Livy's magnum opus. We read of the various conflicts with Veii and the eventual destruction of Veii. The most important person in the defeat of Veii is the Roman dictator, Camillus. The battle is swift in Livy, but the religious aura is elaborate. This is Livy's focus.
Before Camillus gives battle against the Veians, he vows to celebrate games to the gods, to dedicate a temple of Mater Matuta, to offer a tithe of all the spoils of war to Pythian Apollo, and to call Queen Juno to come and dwell in Rome.
Livy suggests that this religious commitment of piety ensured Roman victory. Livy even introduces a tale to show the favor of the gods towards the Romans. As the king of Veii was sacrificing, a few Roman soldiers overhear a soothsayer declaring that whoever cuts up the inwards of the sacrifice will obtain victory. As the tale progresses, the Roman soldiers, who were in a mine under the temple of Juno were able to get that victim and carry it to Camillus.Roman victory takes place swiftly, as could be expected. Divine favor is underlined.
More importantly, we are able to see divine favor in the evocatio of Juno. Livy states that when one of the Roman youth jocularly asked Juno whether she will go, many stated that Juno nodded in assent, and later it was added that some even heard her say that she was willing. Finally, Livy states that the ease with which the Romans were able to move Juno to Rome was further proof that she willingly went to Rome into her new dwelling place.
In short, Camillus' religious piety was the most important factor in the defeat of Veii.