I looks like you plan to go the “normal”/old way of wood tools before stone tools. But it always kind of bugged me that you could make tools out of wood which could break stone.
So I’d have suggestion for this “tier” of tools. Why not go directly to stone?
But how do you make the stone tools then? Just as our ancestors would have. By hammering two rocks together. Also known as knapping. Find two rocks laying around on the floor and knap the tools you need.
(Btw this idea isn’t really mine, a MC modpack named Terrafirmacraft already did this to make MC a bit more realistic.)