Somewhere, maybe here, I have seen T-nuts used for saddle attachments. Quite a range in sizes available 8-32, 1/4 20 tpi, etc. If you have access to the underside of the tree without too much work, they won't pull out. Lots more holding power than screws. Can enlarge the damaged screw hole to insert them.
Just scrolled back through to the beginning of this thread and see this is what @Hildebrand suggested @Mendosa in the 2nd post!