Yes, taken isolation it is not certain that Jesus has in mind a passage such as Jeremiah 14:11-16. But when we take the whole context into account, including the clear allusion to Micah 7:6, and other statements in the Gospels, a more or less literal interpretation of Jesus’ words seems inescapable.

To start with, the surrounding passage in Matthew 10 suggests that Jesus is thinking of violent circumstances: the disciples are sent out as sheep in the midst of wolves, they will be flogged in the synagogues, brother will deliver brother over to death, children will have their parents put to death, but they will not have gone through all the towns of Israel before the Son of man comes (10:16-23); and because Jesus brings a sword, because he has come to set family members against each other, his followers must be willing to lose their lives for his sake.

This may be as much as Jesus is saying, but in Micah family strife (Jesus virtually quotes the text) is a symptom of Israel’s sin (7:1-7); it is one of the reasons why YHWH will ‘strike you with a grievous blow, making you desolate because of your sins’ (Mic. 6:13). That ‘grievous blow’ is war. Given the fact that the war against Rome broke out within a generation after Jesus’ death, resulting in the destruction of Jerusalem and massive loss of life, why would we want to avoid what seems to me the very natural conclusion that he spoke in a prophetic idiom of impending violence that would be the final judgment of YHWH on his people because of their sins?

Why do we think that Jesus was only interested in spiritual things and not willing or able to foresee the destruction of his people and interpret that event through the eyes of the Old Testament prophets (notice the next paragraph: ‘The one who receives a prophet because he is a prophet will receive a prophet’s reward’: Matt. 10:41)? Or why would we imagine that his Jewish hearers, thoroughly versed in the language of the prophets, painfully aware of the nation’s history of violent conflict and oppression, facing the daily humiliation and dangers of living under Roman rule (cf. Lk. 13:11), would have supposed that he was only speaking metaphorically of a sword?

The theme of devastating judgment runs right through the Gospels, from John the Baptist’s warning that the axe is already laid to the root of the trees, that any tree that does not bear good fruit will be thrown cut down and thrown into the fire (Matt. 3:10), to Jesus’ parable about a king who sends his troops to destroy the city of those who refuse to come to his banquet (Matt. 22:7), to his quite literal prophecy of the event in Luke:

And when he drew near and saw the city, he wept over it, saying, "Would that you, even you, had known on this day the things that make for peace! But now they are hidden from your eyes. For the days will come upon you, when your enemies will set up a barricade around you and surround you and hem you in on every side and tear you down to the ground, you and your children within you. And they will not leave one stone upon another in you, because you did not know  the time of your visitation." (Lk. 19:41-44)

In response to your suggestion that he is thinking of a division of the body politic, I would point out that this would be a very Greek metaphor, quite uncharacteristic of Jesus’ thought. It is not, as far as I am aware, found in the Jewish scriptures. There is, moreover, no mention of a ‘body’ in the passage. What we have quite unmistakably is the language of Old Testament prophecy. So I think we have to ask first whether we can make literary and historical sense of Jesus’ words against that obviously relevant background.

At base in the entire thread is the nagging question of how can we
square a God which countenances the wholesale annihilation of races of
human beings with a caring, loving, merciful, compassionate, empathetic
 God.

You presume that this "nagging question" can be squared.  Maybe it cannot.  Maybe the question has nothing to do with the nature of our God, which we can do little about, but has everything to do with your faith in that God. 

Maybe the God of Israel is Love and Fury in One.  Perhaps the Unity of God is Plural and Complex.