I agree 100% that "mental health" carries stigma, and I salute you for trying to find a new term.

I'm iffy on "brain health" for two reasons:
_____ 1. It's very broad, encompassing things like memory loss and cognitive function.
_____ 2. As someone whose depression and anxiety stem from having been the target, from toddlerhood on (if not earlier), of a narcissistic mother and two bullying (and possibly narcissistic) older sibs, I'm fighting against the notion that my challenges are the result of some inherent brain abnormality rather than a toxic environment. To me, "brain health" lets the toxic relatives who did so much damage off the hook.
. . . . . (I acknowledge that our experiences carve neural pathways and therefore my family did affect my brain, but to me "brain health" puts the onus on the person whose brain it is rather than on those who did the damage.)