The axe is just one of many weapons not allowed to a druid, and as druid if one is to gather wood in the forest, how do you do that when the forests are clean of any of the usual forest debris.
There would be a way around the whole axe requiring martial weapon feat or monk class, why not have sickles specifically for the purpose of felling trees. Instead of the usual hammer that is required for forging weapons. Have a stone club for that particular purpose.
You can hammer copper with a rock, though having access to granite would allow you to forge weapons of harder metals, but still you would not be allowed hammer materials of iron and beyond.
Like an icicle which can be lethal if you happened to be under one when it falls. Why not have a druid be able to create crystal blades instead of the usual metal ones. A druid would still need gem crafting tools and gem cutters chisel. And they, like the other classes would still have to go into the various mines for the materails they seek, but not for metals, but for various crystals.
An obsidian blade would have hard time cutting branches from a tree, but could be used for arrow heads, which would allow a druid to use a bow, if they had that proficiency without feeling guilty.
The various gem types could be the determination of the potency of the blade.
A diamond blade could fell a tree and be of equal value to a blade of adamantite.
Personally, I think druids need access to more natural weapons.
