I know this is Jake's classpect but your explanations are so EASY to understand I love them!! So, what kind of powers and what role would a Page of Hope do/fulfill?
Jake served himself hope quite a lot- Believing what his friends told him to not have to face drama. Having Jane and Dirk court him was too much for him and so instead he believed Jane’s awkward little ‘no I don’t like you THAT way!’ and enforced it to the point it made Jane pretty much a go-to to ramble about his relationship issues- Making her serve him in that sense.
Similarly, when he makes Brain Ghost Dirk real, he’s serving his friends in the battle through Hope, believing enough in something to make it real and an asset for the fight, while also serving himself a little fantasy of Dirk being his protector, his Knight, and calling him his Boyfriend.
And the Hope Field, well. That’s Jake at his best, most relevant and most powerful, channelling the powerful forces of Hope to nullify any incoming attacks and blast Jade away. Shame it didn’t last too long.