The story of the Syro-Phoenician woman (Matthew 15 & Mark 7) is interesting in that it specifically is addressing the question of whether or not Jesus was sent to the Jews only, or to all mankind.
A few backdrops
In Genesis 12, God tells Abraham that he will make of Abraham a great nation (obvious assumption = Israel), but more importantly, that all nations will be blessed through him.
In Isaiah 49:6, it says of the Messiah that "It is too small a thing for you to be my servant to restore the tribes of Jacob and bring back those of Israel I have kept. I will also make you a light for the Gentiles, that you may bring my salvation to the ends of the earth."
That said, the Jews had a definite ethnocentric view.
They were the "Ha'am Ha'aretz" - the people of the Land, and that was what was most important to them. In Jeremiah 7 (NIV), we see them trusting in that:
This is what the LORD Almighty, the God of Israel, says: Reform your ways and your actions, and I will let you live in this place. 4 Do not trust in deceptive words and say, “This is the temple of the LORD, the temple of the LORD, the temple of the LORD!” 5 If you really change your ways and your actions and deal with each other justly, 6 if you do not oppress the foreigner, the fatherless or the widow and do not shed innocent blood in this place, and if you do not follow other gods to your own harm, 7 then I will let you live in this place, in the land I gave your ancestors for ever and ever. 8 But look, you are trusting in deceptive words that are worthless
And, in Jesus' day, the Pharisees, for example, thought they were saved because of their birth. Jesus rebukes them specifically, saying (Luke 3:8):
Prove by the way you live that you have repented of your sins and turned to God. Don't just say to each other, 'We're safe, for we are descendants of Abraham.' That means nothing, for I tell you, God can create children of Abraham from these very stones
In answering the Syro-Phoenician woman, Jesus may have been being sarcastic, saying to that woman, "It isn't right to feed the children's bread to the dogs." What Jesus may have been saying sarcastically could very well have been what the Pharisees really would have thought.
When the woman answers, "Even the dogs eat the crumbs that fall from the table," I'm convinced that Jesus laughed. In any event, he would have understood that she got it - that Jesus really was more than just a Hebrew thing.
Interestingly enough, Jesus heals most of the Jews in person, but there are two notable "healings at a distance." Both this woman's daughter and the Gentile centurion did not need Jesus to be there when he healed. In contrast, Jairus just assumed Jesus had to be in the presence.
The point is this - Jesus may or may not have thought that he was sent to the Jews, but the Jews were pretty clear that they expected the Messiah to be strictly theirs. Whether or not that was God's plan was a completely different story.