As the others have said, it is done for operating flexibility.
However, different aircraft designs have the compromise at different points. I think it's fair to say that a Warrior has the compromise rather a long way down, and while it was probably quite a flexible design when it was new 50+ years ago, when the average Yank or Brit male was say 60kg as compared to 100kg today, it is nowadays only a 2-seater if going on any decent trip.
A PA28-140 with four average adults is likely to be overloaded unless it is carrying so little fuel that the fuel level is way below inspectable level (which itself is a dim way to do things) so a large percentage of them are routinely overloaded, and people get away with it because they are coming off a long runway.
PPL training has a lot to answer for. In 2000 (UK), I did W&B but density altitude was never covered, I never saw the handbook for anything that was being flown, and certainly never saw a proper takeoff performance chart of the sort that takes into account elevation, temperature, weight, runway-end obstacle clearance, etc.
I am not suggesting this accident was caused solely by overloading and density altitude though, for the reasons I gave earlier. The aircraft was already airborne, apparently out of ground effect, so some loss of engine power seems a likely possibility. This may or may not be evident in the AAIB report; they can't find what they can't find... I have read many of those reports and while their speculation is educated it is often just that... speculation.