yep, i've seen that example where a mouse could break a climbing rope under certain, ideal conditions....if i recall....as you approach horizontal then the force tends to infinity in these conditions
here though, the rope isn't horizontal, it grabs bark, the stick grabs bark, and the stick is levered into the tree and the friction between the stick and tree (or the teeth digging in) holds a lot of weight
if a stick broke free, i'd be fine unless i was going around a branch (the only time I don't have lanyard and tether in play at the same time)...if it broke right then then i'd have either my lanyard or tether to help me out