[hakuseki]

A small latitude difference does not imply that you can ignore curvature.

I am just guessing, but perhaps your intuition is that the shortest path should lie along a line of latitude. An easy-to-see counterexample would be two points close to the north pole, but with 180 degrees of longitude separating them. In this case the shortest path actually goes through the north pole, rather than around it.

You can also generalize this say that within the northern hemisphere, the shortest path between two points will curve (when viewed on a flat map) towards the north pole.