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It seems my plant is probably Epiphyllum oxypetalum, which is unusual compared to other Epiphyllum sp. in that it is supposed to be a large lanky plant.
So the long branches with nothing but roots are normal, they're for connecting the plant to tree trunks which certainly makes sense. The species does grow long stems which only sprout "leaves" once they reach a certain length.
A healthy Epiphyllum oxypetalum. I guess the flowers make it worth it.
(Photo by Gagan, used under CC license from commons.wikimedia.org
I suppose going forward I should repot, give it more light (full morning to dappled afternoon), and start staking the branches