Jesus, a racist?

Today, we read the Matthean version of the story of Canaanite woman whom Jesus implied 'a dog'. To the ears of modern men, Jesus' statement was very racist. It is embarrassing for modern Christians to defend this racist statement for their beloved Jesus. Isn't Jesus always a compassionate Saviour, the champion for the poor, the defender of the underdogs? Traditional apologists would defend Jesus' statement that Jesus was testing the faith of the woman. True, Jesus praised her faith in the end (Matthew 15:28). But I wonder if the audience of the first century would feel it this way.
It all began when Jesus clashed with the Pharisees. The Pharisees started criticizing Jesus' disciples for not washing their hands according to the manners prescribed by the tradition of the elders (Matthew 15:2). Jesus retorted that they did not follow God's commandment to honour their parents and quoted Isaiah to expose their hypocrisy. Predictably, the Pharisees were offended (Matthew 15:12). Yet, Jesus did not stop and continued to criticize their false teachings.
Let them alone; they are blind guides. And if a blind man leads a blind man, both will fall into a pit (Matthew 15:14).
After that, Jesus retired to the region of Tyre and Sidon (Matthew 15:21), a Gentile region.
Now, a certain Canaanite woman came (from behind) to beg Jesus to exorcize a demon from her daughter (Matthew 15:22). This Matthean version has the woman calling Jesus, the Son of David. Moreover, this version also provides an occasion for Jesus to explain to his disciples why he did not show his mercy.
He answered, "I was sent only to the lost sheep of the house of Israel." (Matthew 15:24)
The woman in the Marcan version was a Greek and Mark does not have this extra bit of information, making Jesus more blatantly racist.
Back to Matthew, the woman knelt before Jesus to beg him. Now, here comes the embarrassing statement.
And he answered, "It is not fair to take the children's bread and throw it to the dogs." (Matthew 15:26)
We have no way to know exactly what Jesus spoke in Aramaic. But the Greek word for 'the dogs' τοῖς κυναρίοις is neuter in gender, not feminine. Still, for modern ears, the language is offensive because Jesus degraded the Canaanite to the status of dogs! The reason Jesus gave for refusing to help the Canaanite woman was that she was not an Israelite. Worse, Jesus compared her to a dog!
But I suspect that this debasing connotation is probably a modern phenomenon.
When we turn to the Old Testament, we will see some interesting relations between man and dogs.
You shall be men consecrated to me; therefore you shall not eat any flesh that is torn by beasts in the field; you shall cast it to the dogs (Exodus 22:31).
So, dogs are pretty useful. They work like scavengers for man. When people die as a punishment from God, dogs help eat up their corpses to clean up the places (1 Kings 14:11, 16:4, 21:23, Psalm 22:16, Jeremiah 15:3). Dog licked blood, the blood of both Naboth and Ahab (1 Kings 21:19, 22:38, Psalm 68:23). It is interesting to note that God helped Gideon select 300 soldiers who lapped water with their tongues like dogs to fight against tens of thousands of Midianites and Amalekites (Judges 7:5).
Of course, dogs, because of their diet, were despised animals. Goliath was furious to see David come to challenge him with sticks only.
And the Philistine said to David, "Am I a dog, that you come to me with sticks?" And the Philistine cursed David by his gods (1 Samuel 17:43).
You humbled yourself by calling yourself a dog (2 Kings 8:13) or even a dead dog (1 Samuel 24:14, 2 Samuel 9:8). Of course, you (Abishai) can curse your enemy (Shimei) a dead dog (2 Samuel 16:9). By calling people 'dogs' in their faces was (and is) meant to humiliate them.
In Matthew, only two occasions mention 'dogs'. The other one appears earlier.
Do not give dogs what is holy; and do not throw your pearls before swine, lest they trample them under foot and turn to attack you (Matthew 7:6).
Here, dogs and pigs are mentioned in the same breath. They represent people who reject the good news, the generous invitation of God. They are both Jews and Gentiles. Therefore, the use of 'dogs' in this context is not racist. It does suggest people who are ungrateful to God, ignorant of God (Isaiah 56:10) and even hostile to the evangelists.
Jesus did not call the Canaanite 'a dog' directly like Abishai. He was speaking rather objectively, stating some facts of life, though the implication was very strong. Now, it would be insulting only if you admit that you were a dog! Jesus didn't say it to that effect. Here comes the superiority of the woman's answer.
She said, "Yes, Lord, yet even the dogs eat the crumbs that fall from their masters'  table."  (Matthew 15:27)
OK, let's talk objectively about dogs!
Either the master is too generous so that there are leftovers for dogs, or that the children (the Pharisees) dislike your food (salvation). Both ways, dogs win! Am I a dog? That is not the question. The Canaanite woman beat Jesus hands down! (Do you see, in your mind's eye, a dog going off with its tail between its legs? I am only joking!)

My dear Jesus, I still hold you dear in my heart. You are still very compassionate for me. You did not deny the Canaanite woman salvation, but had given her a show time. I pray that I am able to see every obstacle an occasion for me to flex my muscle, to love You more. Amen.