The way how chopping feels is kinda odd, because I wasn't sure if I should time it right.
So I kept simply clicking fast and upgrading it with what I got.
Managed to enter the inside of that tree and get to the mine minigame.
When I click anywhere in the inventory, I can still see the playable character swing the axe.
Will the mine minigame also have a go-up button?
There's some interesting stuff here.