I think Emma's role in current X-books is her trying to be a leader where the other leaders are failing in the mutant community. Xavier is off-world with bird wife, Rogue's team is just living down in Louisiana, Scott and Magneto's team is just scraping by in a factory.
Mutants have very little community right now, especially for the concept of younger mutants who do not have the luck of being in New Orleans or involved with 3K enough to be of interest to Scott dealing with them. Everyone is too busy to actually deal with the fact that the loss of Krakoa means the loss of a common mutant community. NYX was cancelled before it could serve as that concept.
The Hellfire Vigil was Emma's idea.
Teaching the kids from Exceptional was her idea.
Graymatter Lane is her idea, a safe place for mutants anywhere in the world to find community. I don't think she's power hungry, I think she (and Kate) are the only ones willing right now to reach out to ANY mutant who needs or wants community. And she'll be damned if someone like Scott is going to get in the way of her trying to help mutants when no one else is. Even Magneto is onboard with the school concept, even when Scott tried to raze the mind palace to the ground.
I don't think Scott is power hungry either, but he's more like Xavier right now than he will ever admit.