I think this is because the edge needs to hit a certain collider for the player to reach for it, so, say, if shadows box collider hits the edge of a platform before the edge detection trigger does, then it treats it as a normal ground tag, so, if you jump before the edge reaches the trigger, the player will jump as if he's on the ground.
This is actually intentional. There are ways, if you're really good with the movement, where you can exploit this to skip having to climb the edges, which could save you several seconds, which is really big in speedrunning, which is a big target demographic for this game. But I can see how that could be annoying or not very obvious.
Would you say you'd prefer for the player to always grab the ledge when you hit one, say, set jumping and moving to false if the edge is within a certain range, or would you think it be sufficient to bring attention to this? I do actually plan to implement a tutorial in the near future, so, if the latter then I could make this obvious in the tutorial level. I can do either option, but I would like to get some more opinions on this matter as well.
If you'd like to play the game further, try waiting a moment when you hit the edge of the platform if the player doesn't immediately grab it. The trigger for the player's edge detection is near his face or a bit higher, see if that lets you get a bit further until I've come to a proper solution for this.