I went to the woodworking table as a level 10 in handiness skill and there is no option for the wooden throne. So I made the toilet hoping that would complete the mission, I know it’s not the same but I tried it. Of course it didn’t work.
Also, I went to the cave and in my game it’s already broken into, so I explored it and it didn’t check off the mission from the list.
Edit: my game is updated to the latest version and I have no mods.
I had the same problem. You have to claim the wooden throne from the event first and then you can make it on the woodworking table.
