I tend to read the passage of the Syro-Phoenician woman a little more strongly. She didn't just change Jesus' mind: she corrected him.

In their exchange, Jesus appears cruel.

25 Then the woman came over to Jesus and bowed before him. She said, “Lord, help me!”

26 He answered her with this saying: “It is not right to take the children’s bread and give it to the dogs.”

27 The woman said, “Yes, Lord, but even the dogs eat the pieces of food that fall from their master’s table.”

In the compressed, external narrative of the Gospels we are not privy to many deliberations or realizations within Jesus, but one wonders whether this woman did not expand Jesus' vision of his ministry with her cheeky/desperate pushback.