I found this to be helpful and open up the story as I had never seen before. Hope this Bible Commentary answers questions.
MacLaren’s Expositions
Mark
CHILDREN AND LITTLE DOGS
Mark 7:24 – Mark 7:30.
(Excerpt)
“Our Lord desired to withdraw from the excited crowds who were flocking after Him as a mere miracle-worker and from the hostile espionage of emissaries of the Pharisees, ‘which had come from Jerusalem.’ Therefore He sought seclusion in heathen territory. He, too, knew the need of quiet, and felt the longing to plunge into privacy, to escape for a time from the pressure of admirers and of foes, and to go where no man knew Him. How near to us that brings Him! And how the remembrance of it helps to explain His demeanour to the Syrophcenician woman, so unlike His usual tone! Naturally the presence of Jesus leaked out, and perhaps the very effort to avoid notice attracted it. Rumour would have carried His name across the border, and the tidings of His being among them would stir hope in some hearts that felt the need of His help. Of such was this woman, whom Mark describes first, generally, as a ‘Greek’ {that is, a Gentile}, and then particularly as ‘a Syrophcenician by race’; that is, one of that branch of the Phoenician race who inhabited maritime Syria, in contradistinction from the other branch inhabiting North-eastern Africa, Carthage, and its neighbourhood. Her deep need made her bold and persistent, as we learn in detail from Matthew, who is in this narrative more graphic than Mark. He tells us that she attacked Jesus in the way, and followed Him, pouring out her loud petitions, to the annoyance of the disciples. They thought that they were carrying out His wish for privacy in suggesting that it would be best to ‘send her away’ with her prayer granted, and so stop her ‘crying after us,’ which might raise a crowd, and defeat the wish. We owe to Matthew the further facts of the woman’s recognition of Jesus as ‘the Son of David,’ and of the strange ignoring of her cries, and of His answer to the disciples’ suggestion, in which He limited His mission to Israel, and so explained to them His silence to her. Mark omits all these points, and focuses all the light on the two things-Christ’s strange and apparently harsh refusal, and the woman’s answer, which won her cause.
Certainly our Lord’s words are startlingly unlike Him, and as startlingly like the Jewish pride of race and contempt for Gentiles. But that the woman did not take them so is clear; and that was not due only to her faith, but to something in Him which gave her faith a foothold. We are surely not to suppose that she drew from His words an inference which He did not perceive in them, and that He was, as some commentators put it, ‘caught in His own words.’ Mark alone gives us the first clause of Christ’s answer to the woman’s petition: ‘Let the children first be filled.’ And that ‘first’ distinctly says that their prerogative is priority, not monopoly. If there is a ‘first,’ there will follow a second. The very image of the great house in which the children sit at the table, and the ‘little dogs’ are in the room, implies that children and dogs are part of one household; and Jesus meant by it just what the woman found in it,-the assurance that the meal-time for the dogs would come when the children had done. That is but a picturesque way of stating the method of divine revelation through the medium of the chosen people, and the objections to Christ’s words come at last to be objections to the ‘committing’ of the ‘oracles of God’ to the Jewish race; that is to say, objections to the only possible way by which a historical revelation could be given. It must have personal mediums, a place and a sequence. It must prepare fit vehicles for itself and gradually grow in clearness and contents. And all this is just to say that revelation for the world must be first the possession of a race. The fire must have a hearth on which it can be kindled and burn, till it is sufficient to bear being carried thence.”