I've stumbled across a proof of geodesic stability in hyperbolic space, located in the following blog post:
Maybe it would be wiser to comment there directly, but I figured I would ask here first. Is the proof of lemma 5 valid? I am really not convinced of it. I know where to find a proof in the setting of $\delta$-hyperbolic spaces, but it would be nice to use this one since it's quite simple in comparison.