(This'll be the last time I talk about this character for awhile. Yes, I am hating on this character, again, but I don't care).
If you were to remove Hazel from the story, nothing really changes (except a few things). I mean the guy had absolutely no chemistry with any of his cohorts. We have no idea what his thoughts were on Cinder, Watts, or Tyrian. Or what his dynamic was like with Gretchen. He wasn't even a good foil for the heroes. Plus his motivations for joining the villains are extremely stupid and hypocritical, and his redemption is as complex as flipping on a light switch.
Mini tangent aside, if you were to simply remove Hazel from the story outright, a lot of the story beats could still play out the same. Like Oscar at the train station and him getting upset at the broken ticket dispenser. You could remove the scene. I know it's supposed to show Oscar getting flustered over something so trivial and Hazel tells him not to let something so small stop him. But you still could remove it because we have other scenes of Oscar getting flustered like when he's "training" with Ruby and not being able to activate his aura. During the god awful Battle at Haven, if Hazel was removed, you could have Emerald and Mercury fighting Nora, and Ren. While Oscar fights Lionheart.
Then there's Volume 8. Some would ask, "If Hazel's removed from the story, then how's Oscar gonna free himself, and how are the heroes going to escape from Salem?" Have Mercury act as Oscar's jailer/torturer. I mean, it would give him something to do. Because there was really no need for Mercury to be in Volume 8. Then when he's called to go to Vacuo with Tyrian. Have Salem make Emerald be the farmhand's jailer. Due to her fear of Salem, Emerald is forced to oblige.
Here we could have Oscar give Emerald the password to the Lamp. However, some might ask, "how's Emerald gonna disguise Oscar if Hazel isn't in it?" Easy, she uses her semblance to make him invisible. Her superpower is mind manipulation and she was able to dupe Salem into thinking Oscar was Hazel. So why not make her think Oscar's invisible.
But how are the heroes gonna escape without Hazel's final stand against Salem? You still could have Oscar activating the cane's stored up kinetic energy a.k.a. the cane nuke at Salem. But how is he gonna activate the "cane nuke" if Hazel isn't there to give him his cane back? Have Emerald give him back his cane before they're found out by Salem. So when the heroes are pinned down. Oscar could use the stored up Kinetic energy to form a force field around himself (and the others) to nuke Salem and her whale.
In conclusion, Hazel's involvement in the overall narrative is pretty much unnecessary. You still could achieve the same results with the main story. You just gotta tweak a few things here and there.