The hat is not a requirement to fulfill it or not.
The hat is a prompt that lets you know you have a task. If you had a task that is not based on a timer and you just sat there… of course you didn’t get anything done. You actually have to fulfill these to finish they will not go away on their own. You also will not get credit if you fail to do this
If the task was on a timer, you should have gotten credit and finished weekly regardless if you helped or not. If you didn’t then you need to leave the area and come back because something didn’t go right. Sometimes if you kill the meat parts too fast it will bug.
The hat is just a speed buff. You do not have to press it. Pressing it doesn’t help or hurt