You can recover a distance metric from the cosine similarity of unit (i.e. normalized) vectors by taking their Euclidean distance, which can be written as basically a square root of the complement of cosine similarity. Or you can just take the complement and forget the square root, which isn't technically a distance metric but might be good enough. Or you can invert cosine similarity to get angular distance, which is a true distance metric but might be too expensive.
Both chord length and arc length can be good ones. For some purposes versine (1 - cosine, sometimes called the "normal distance") and the half-tangent of arc length (i.e. the distance to one point when stereographically projected so the center of projection is the other point) are also useful types of quasi-distance. https://davidhestenes.net/geocalc/pdf/CompGeom-ch3.pdf
