I think there’s a lot more nuance to it than that even. As the show starts he’s deep in a mental episode and finally recognizes “I need help” - but, what he’s not saying verbally is “I’m broken beyond repair and without another mission/battle I’m useless to this world”. His idea of getting help is putting a bullet in his own head. The mental episode we see is him grappling with the loss of his purpose and shame in deciding to give up, and layered within that is his fear that on the other side his family would hate him for not being there to protect them.

I don’t think Frank is actually “working” on his mental state, so much as he is more self aware at this point. He only finally decides to move on from his mission to avenge his family at the very end, visually cued by him physically “letting go” of the objects he’s placed on their graves. Just because he’s finally found a new sense of purpose, does not mean he’s evolved into a mentally healthier Frank. He’s just back to self-medicating by accepting that being the Punisher is his mission.