Why does Revelation say Jesus will shepherd with a rod of iron when Psalm 2 says he will break with it?

Because John is quoting the Septuagint, and the Septuagint read the first verb as "shepherd" for reasons the rest of the Greek Old Testament makes obvious.

Here is the verse from Psalm 2:9 as the Masoretes pointed it:

"You shall break them with a rod of iron; like a potter's vessel you shall shatter them." — Psalm 2:9 (MT)

And here it is as the Septuagint rendered it:

"You shall shepherd them with a rod of iron; like a potter's vessel you shall shatter them." — Psalm 2:9 (LXX)

Only the first verb changes. The unpointed Hebrew letters are ת-ר-ע-ם. With one set of vowels, the word is tero'em from ra'a' (רָעַע, H7489, "to break"). With another set of vowels, the same letters spell tir'em from ra'ah (רָעָה, H7462, "to shepherd"). The two roots share two of their three letters, and the suffixed form collapses the difference. The vowels were added to the written text around a thousand years after the Septuagint translation was made, so the earlier translators were reading bare consonants.

The Greek translator's choice was not random. Across the whole Septuagint, the Greek verb poimainō (ποιμαίνω, G4165, "to shepherd") renders the Hebrew ra'ah fifty-one times — and it never renders ra'a' ("to break") at all. The translator was reading the verb as "shepherd" and translating it the way he always did.

John of Patmos quotes Psalm 2:9 three times (Rev 2:27, 12:5, 19:15), and every time he takes the Septuagint's shepherd-verb:

"And he will shepherd them with a rod of iron." — Revelation 19:15

The striking thing is that John had the other option sitting right next to him in his cultural moment. A Jewish messianic text from the century before Christ — Psalms of Solomon 17:24 — quotes Psalm 2:9 and uses syntripsai (συντρῖψαι, G4937, "to shatter") instead. That was the dominant Second Temple reading. John did not take it. Three times, at the three structural hinges of Revelation, he wrote poimanei — "he will shepherd."

Why? Because the canon had already welded the shepherd-king image to the Davidic throne. Yahweh tells David, "you shall shepherd my people Israel" (2 Sam 5:2). Micah 5 names the coming Bethlehem ruler as the one who will shepherd his flock. Ezekiel 34:23 promises "one shepherd, my servant David." The Messiah was the shepherd-king before he was ever the conqueror, and the rod he carried was a shepherd's staff before it was ever a war-club.

The rod does both. That's the point. For the full argument — and for how Daniel 2's stone fits into the same picture — see The Rod of Iron.