I'm not arguing against the idea that you should be cautious and definitely make sure you've done the calculations, (and check in on anyone you see doing the same) but it's not like 4 adults in a 172 is a guaranteed fatal.
Yes, you're absolutely right. The comments I had made were under the assumption that the accident was caused by an overweight issue. Given the NTSB's statement, it looks like this has more to do with the flap setting. I've flown with 4 people in a C172SP in the past, with enough fuel to put us over by literally a pound or two. Flights in the C172M are doable if you're overweight, but I wouldn't be one to do that.
I fly an identical model 172M, and the checklists we use call for the flaps to be put all the way down during the preflight, as do all C172s. The second item we perform on the Engine Start Checklist after the engine is started is to raise the flaps. During the Before Takeoff Checklist, the flaps are checked again. This could be a case of simply not following the checklist.
I still believe the student knew he was overweight before he departed. During the failing climb out, it seems he jumped to the immediate conclusion that he was having a problem due to this overweight condition, which was perhaps on his conscience as a new pilot. I doubt he realized at any point that the flaps were the issue.