Jesus giving answers he doesn't mean- 7:24-30

This woman immediately (euthus) upon hearing of Jesus arrival, came and fell at his feet. How did Jesus know the woman was not a Jew. They weren't in Jewish territory- would there still have been some Jews that lived in the region- I imagine so. Was it the fact she spoke in Greek rather than aramaic? Or did he ask more questions than we have in the account. Somehow it has come up that she was born in Syrian Phoenicia- so perhaps not far away or a bit further up the coast. Apparently there is much that is irregular in her approach in Jewish culture- and in Jesus response (that he is willing to make a response). This is probably a passage where this sort of context is crucial- The situation is not as harsh as it reads. Jesus was sent to the children of Israel- there was a long history intertwined with Yahweh- they were his chosen people, in a way no other people were. But in another cultural blindspot for the pharisees- the reason for this was so that the 'dogs could eat the food from the children's table'. So Jesus here gives a good rabinical answer- that he of course does not mean. He knows the privilege of the Jewish people. But there are other sheep not of this sheep pen- and they will listen to his voice. The 'Jews' ending up killing Jesus- which contrasts starkly with the humility of this woman- I will take whatever scraps I can get.