The goal of the weird route is basically forcing Noelle's ice magic to exist in the light world. She does this by you slowly blurring the line between the dark world (dream) and the light world (reality) within her head.

When you give her the ThornRing, you literally turn her room into a pocket dark world because she fully believes what's happening is real. That’s what makes it possible for the ThornRing to show up in the light world.

The pain from the ThornRing is important because it's like a 24/7 reminder for Noelle to never stop blurring the line between both worlds.

Next time you meet her and tell her to cast snowgrave, would she believe that it's not possible?