Jeremiah 31.10, 11-12ab, 13 Resp. 10d
The first reading was written before the exile had really begun, but Jeremiah is already prophesying about the glorious return, not of the whole nation, but of the remnant. This remnant of Israel was a people who had been purified by suffering and whose faith was deeply informed by Scripture and obedience to God. This does not mean that they were perfect. Jesus often complains about the Pharisees, but the Jews, when they went wrong, did so in the direction of hypocritical obedience.
So it should not surprise us when Jesus says that he has been sent only to the lost sheep of the house of Israel. He is saying that he was sent only to those who were ready. The Jews might kill him for claiming to be God, but the other cultures would have just been confused. They would have called him a magician. They would have thought that he was one of their many gods walking on earth. Other cities might have bowed down in worship, indeed Jesus says they would have, but they would not have understood what they were doing. No other culture had been prepared in the way that the Jews had been.
So the woman asks for her daughter to be freed from the demon, and Jesus ignores her. He did not come to earth to do magic tricks. His miracles are always signs of his identity. Why would you make a sign for someone who cannot read it? The miracles of Jesus only convince some of the Jews, but even a simple Jew like Simon the fisherman had enough cultural heritage to recognize the Savior of the World just from a big catch of fish. The woman has not had the same preparation, so she cannot be expected to understand the signs.
Then she says, “Please Lord, for even the dogs eat the scraps.” It is usually assumed that she means that one little miracle could be considered a scrap, and perhaps she did, but she may also have meant that she has lived near the Jews long enough to pick up a few scraps of their faith. Certainly, at least, enough to know who Jesus is. She is not just a desperate mother who is reaching out to every possible solution for her daughter’s affliction; she is a mother who knows that this man is her only hope, and she is unwilling to accepts no for an answer when she believes that he can do it.