Hey,
Pathfinding has been the same in the client for a while. Here is an 'official' documentation
https://oldschool.runescape.wiki/w/Pathfinding So to be clear, they actually use Breadth first search and not Depth first. It is a rather 'greedy' approach but will give you the shortest path. A* is weighted and uses a heuristic (manhattan distance/euclidean/chebysnev etc). Your client will for sure have code to demonstrate how this handled. I'm a bit worried because you probably should not need to pay for this. Also when you say it has edge cases, can you elaborate? Show us a scenario or something I am rather curious. Are any edge cases occurring whilst in combat? If so, you must note that the calculations you perform in order to move your player to the tile you need to be on in order to attack your opponent needs to be considered. BFS is still used however there are checks and other things to consider when choosing the tile you need to move to.
I have both implemented, BFS and AStar. I have compared the path constructed in many scenarios. The path's differ in certain scenarios and in other scenarios it does not. There are also no edge cases my implementation has caused the path finder to 'fail' or act 'incorrect' or seem 'unjustified' for the path it has calculated. It would also be good to know which scenarios you are mentioning.