I think half the problem here is that 越える and 越す are totally on the verge of the 自動詞／他動詞 definition, to the point I look to a sentence and don’t know if it should be considered one or another.
Why so? For example, 山を越す。It’s very easy to say it’s a 他動詞、there is a を and all. But the thing is Japanese also uses を for movement verbs.
For example, 公園を歩く。 No one would argue that 歩く is 自動詞, right? So why exactly 越す would be 他動詞 in 山を越す? It is describing a movement after all. And it also doesn’t imply a direct action towards the mountain, as 山を登る would.
I’m not saying the answer is one or another, though. I feel like I don’t have enough linguistics knowledge to answer that.