Your comment made me run a few tests.
You are supposed to get a phonecall that initiates the Wood cut task, the phonecalls that come in are random.
However! In checking the game I found an issue where the wood cutting phonecall rng would never reach the correct value after the other phonecalls had come in. So, if in a playthrough you had the spooky giggle call and the old person already, it COULD become impossible for the wood cutting task to initiate in that same playthrough.
This is now fixed :) Good luck!