In everything I've read that involves her interaction with law enforcement, it seems like "mental health" has been used as a justification. In the case of the Moab stop, the report emphasizes her mental health because they were preemptively justifying not arresting her. They said it was a "mental health crisis" more than DV because they had concerns about their compliance with the statute.
In the missing persons report, I think that getting resources allocated to her case hinged on it being "special" in some way. The route they decided to go was "mental illness/mental health". Remember that her mother called the one LEO that helped her "her angel" -- I would bet that part of being an angel was finding a way to cut through the red tape.
My own opinion is that Gabby was in no way mentally ill. Maybe she had anxiety, many of us do, but I think that "mental health" and "mental illness" were bandied about as a way to justify LE actions, and not because she actually had serious mental health issues that went beyond the normal problems that almost everyone has.