Both good considerations! This approach can be used to build a kernel that measures the similarity of two items with respect to the labels learned by the forest

Both good considerations! This approach can be used to build a kernel that measures the similarity of two items with respect to the labels learned by the forest. I don't doubt that the specific algorithm used to grow the forest. Luckily, any kernel made from any forest will have a positive semi-definite Gram matrix. But that doesn't mean it'll be a good one. I would summarize this approach as "learning a kernel that compares items w.r.t. the labels learned by the tree."