Let me add my 2 cent here.
You are right about the usefulness of the fatigue report.
However you also know how this company constantly threatens the employees. Maybe you guys from the flight deck don't really feel it as much as CC do, but I assure you that among the CC community there is a lot of fear of the company and few, if any CC trust the management.
Let me explain myself.
When I was CC, I heard the trainers at the training college telling us several times that if you fill a fatigue report, there will be a "thorough investigation to determine the causes of what happened. The company will investigate how many hours of rest you had in between your flights, how was your health, how you used that rest, etc" basically saying that the company will check all the facts and find a way to put the guilt on you. "You should have slept better and you still decided to operate a flight knowing that you weren't fit" so here you go, now the CC is the responsible and will get a nice final warning or worse. This is the message that the company transmits to the CC. Ask around if you think I'm exaggerating.
And as CC I was constantly feeling threatened by the company to not fill these reports. We all know how fair is this region. They can throw you away with no explanation what so ever and by filling the report you are essentially reporting yourself for not getting enough rest. In the eyes of the company it's easier to bully the crew and keep us shout rather than increase the rest hours, everybody knows that. You don't have to be a guru to understand that 5 night turnarounds in 7 days is not healthy and that it causes fatigue. They know and they don't care. I filled one myself after some horrible night TA and a month later a got a 3 lines reply saying that the flight is legal and there is nothing to discuss.
So if actually you guys, the pilots, take the initiative to ask the crew if they are tiered and then offer to be the ones to fill the report and the crew signs or something like this, you will be taking most of the fear off the crew, since now the report is initiated and backed by the captain\FO. And I doubt that the management would go against a captain for reporting the crew was tiered. Try doing this and I assure you, you'll have many more reports filled. The CC just needs to feel more backed up!