NG+ lets you choose to start right before the branch. Doing the entire game over again is pretty trivial anyways, you will one-shot everything and can just disable encounters. It's good to see how the Alice/Ilias interactions differ imo.
The MTL, for whatever reason, includes some save data that lets you do the third route in all your saves, even without doing the other two. You are intended to do both other routes before the third - and you're intended to get a bad ending for trying to do the third route beforehand.
All three routes of part 3 should be done. They're all completely different. Neither the Destroyer of the Heavens or Monster World Judgment routes are fully satisfying, but they're essential for getting the most out of the third route.