A meditation on Matthew 15:24; apparent sarcasm from our Lord's lips

Have you ever noticed where Jesus was (Matthew 15:21) when He said "I was not sent except to the lost sheep of the house of Israel"?

In the coast of Tyre and Sidon, of course--among the Gentiles.  This region had once paid tribute to Solomon, but had never really been part of Israel.  So what was He doing at this point?  Going to the coast of Tyre and Sidon to minister to Jews was like going to Green Bay to find Bears and Viqueens fans, or to Congress to find an honest man.

My hunch is that He was confronting the Pharisees and scribes sent (Matthew 15:1) to monitor His Work.  First, He leads them among the Gentiles, and while ministering to one of them, He implicitly rebukes them by sarcasm.  While the woman responds admirably--and in the vein of what the Pharisees believed about Gentiles--the central point of Jesus' healing of the woman's daughter is that His ministry was to the goyim as well.  As one, I'm grateful......

Addendum: one might also infer from the context that Jesus could be pointing out to the Pharisees how they are departing from their "traditions of the elders"/"Oral Torah" by leading them far among the Gentiles.  He is presenting, implicitly, the Gospel to them.  Along the coast of Tyre and Sidon, they would have traveled approximatly 200 miles from Jerusalem.  Given that they were most likely rather soft-footed scholars, this travel demonstrates they--and most likely the Sanhedrin as well--were taking His ministry very seriously.