Every mission/level has a leaderboard score, but I can't find an obvious way to replay levels to attempt a higher ranking, even from the score menu in Settings. How can I replay specific story levels? Do I have to make a new game and play through again?
You can't return to specific story levels.
To return to a previous level, you'll have to play through the story again. When you finish the game, you can reuse your save for your next playthrough.
Another option: turn off autosave and you can replay the same level by hard-resetting the game after each run.