Paul is quoting Proverbs 25:21-22 here:
If thine enemy hunger, feed him; if he thirst, give him drink;
For so doing thou shalt heap coals of fire upon his head, and the Lord shall reward thee with good.
I think the intent is to shame, rather than soften - but perhaps the shame leads to softening. I agree that the theme is not revenge. It really is repentance. A contemporary Orthodox commentator, Dmitry Royster, explains the passage:
"in so doing," that is, if one acts in accordance with this principle, he "will heap coals of fire upon his enemy's head," a proverbial expression: in this context, it means "put to shame," or "make one painfully conscious of his guilt." It is found also in Psalm 139/140:101, where it denotes God's own retribution on those, Saul and his followers, who were persecuting David in order to kill him, even though he had dealt in a godly way with Saul.2
The Psalm reads:
7O God the Lord, the strength of my salvation, Thou hast covered my head in the day of battle.
8Grant not, O Lord, the desires of the wicked: Further not his wicked device; Lest they exalt themselves.
9As for the head of those that compass me about, Let the mischief of their own lips cover them.
10Let burning coals fall upon them: Let them be cast into the fire; Into deep pits, that they rise not up again.
Archbishop Dmitry continues:
The hope implied by the expression in our present verse is that the enemy's realization of his wrongdoing might lead him to repentance. For it is unthinkable that the man who feeds him and gives him to drink would do these acts of kindness for the purpose of revenge, that is, to see his enemy in spiritual, mental, or physical torment.3
1. Psalm 139 in the Septuagint (from which Paul quotes), Psalm 140 in the Masoretic Text
2. St. Paul's Epistle to the Romans: A Pastoral Commentary (St. Vladimir's Seminary Press, 2008), p.328