The concept of the suppliant was very important in the ancient world, and we in modern times have lost it in the mechanizations of bureaucracy.
A suppliant, or supplicant, is today in English merely one who makes a humble, earnest, and expectantly sincere plea for something from another.
But in the ancient world the idea had a religious connotation attached to it.
Those who refused suppliants were seen as cruel, and invited the wrath of the gods – or of God – upon themselves. Suppliants often acted in desperation, and took olive branches as a sign of their humbled state, sometimes even wearing garments of mourning, throwing themselves at the feet of a ruler, a general, or even an altar, often grasping the garment of the one they sought favor from, and they begged earnestly for the mercy that they wished to receive.
Suppliants in those days are described by the following: Plato, Laws, Book 5 ,Livy, The History of Rome, Book 2 Chapter 14, Livy, The History of Rome, Book 45 Chapter 6, Homer's Odyssey, Book 9
The woman, once having fully – and evidently sincerely - agreed with all of Christ's statements, by all measures of mercy and clemency,
it not actually costing Yahshua anything to grant her wish, He was given little choice but to do so.
He always complied with the cultural norms of the time, and it cost Him nothing, such as when Peter retrieved the coin from the mouth of the fish to pay the strangers' tax.
The woman, while an enemy, was a supplicant who recognized both His kingship and His purpose, and had agreed with Him fully, while prostrating herself at His feet.
Since it was He who also declared that the Wheat and the Tares must live together until the time of the end, He had little choice in the perspective of His Own affirmed righteousness, than to grant her wish as she desired. However
His granting her wish, which was the healing of her daughter, does not mean that she is granted salvation or can claim that which was promised to Israel.