The herem against the peoples
The herem against the peoples of the land (with the campaigns of Joshua and 1 Samuel 15) — divine judgment, hyperbolic ancient war rhetoric, or later ideological projection.
Deuteronomy 20:16-18
16But of the cities of these peoples that Yahweh your God gives you for an inheritance, you shall save alive nothing that breathes;
17but you shall utterly destroy them: the Hittite, the Amorite, the Canaanite, the Perizzite, the Hivite, and the Jebusite, as Yahweh your God has commanded you;
18that they not teach you to follow all their abominations, which they have done for their gods; so would you sin against Yahweh your God.
What each tradition says
Rabbinic / Midrashic
Kohler states that devoting spoils of war to the Deity, whether persons or things, was found among all ancient nations and primitive tribes. This practice is inseparably connected with holy warfare, which claims all booty for the god who leads to victory and in whose honor captured foes and goods are destroyed on the spot.
Kaufmann Kohler (1843-1926), The Jewish Encyclopedia, vol. 2, s.v. 'Ban' (New York: Funk & Wagnalls, 1901-1906) — opening section, on the herem as holy-war practice
Patristic
Augustine says the eternal law requires preserving natural order and forbids transgressing it. Some actions are indifferent, so people are blamed for presumption if they do them without being called upon, but deservedly praised if required. In the order of nature, the act, the agent, and the authority for the action are all of great importance.
Catholic
Aquinas says a distinction was observed regarding hostile cities. Some were far distant and not among those promised; when taken, they killed all the men who fought against God's people, while sparing women and children. In neighboring promised cities, all were ordered slain for former crimes, God sending the Israelites as executor of Divine justice.
Thomas Aquinas (1225-1274); trans. Fathers of the English Dominican Province, Summa Theologica, Prima Secundae (I-II), q. 105, 'The Reason for the Judicial Precepts' (New York: Benziger Bros., 1947) — q. 105, art. 3, reply to objection 4
Orthodox
No sourced position found. Checked the PD Greek/Eastern corpus: John of Damascus (NPNF 2-09) — no mention of Deut. 20:16-18 or the herem; Chrysostom's Acts, Romans and Matthew homilies (NPNF 1-11, 1-10) — no treatment of the ban found; the 1672 Synod of Jerusalem (Confession of Dositheus, archive.org) — no treatment; Origen's allegorical Joshua homilies survive in English only in a modern copyright translation. No PD statement found — an honest gap.
Reformed / Protestant
Calvin argues that charging this command with cruelty wrongly usurps God's authority as judge of all. The objection that God's people were made inhuman, killing every sex and age, is specious. Since God had destined the land for His people, He was free to utterly destroy its former inhabitants so they could possess it.
John Calvin (1509-1564), Commentaries on the Four Last Books of Moses, vol. 2 (Harmony of the Law) (trans. Charles William Bingham, Calvin Translation Society, 1854) — Commentary on Deuteronomy 7:2 (the same 'utterly destroy' statute behind Deut. 20:16-18)
Evangelical
Keil and Delitzsch state that none of these nations was to be left alive. These peoples were laid under the ban, meaning altogether exterminated, so they might not teach the Israelites their abominations and sins.
Carl Friedrich Keil (1807-1888) and Franz Delitzsch (1813-1890), Biblical Commentary on the Old Testament: The Pentateuch, vol. 3 (trans. James Martin) (Edinburgh: T. & T. Clark, 1865) — Commentary on Deuteronomy 20:15-18