I am pretty sure that this:
is incorrect. It should be:
"north": { "uv": [ 1, 9, 15, 16 ], "texture": "#saw", "tintindex": 0 },
Same goes for your other textures. There's a reason for the name definitions at the top.
Though I do see that it says iiwam:item/carpentry_table which does not appear in the files you've shown.