There are lots of "damage-sensing" mechanisms at all scales within the body. These are how your skin knows to grow after an injury, how your immune system detects which cells are infected and need to be killed, and how your cells know which parts of themselves to digest via autophagy, among other things. At the single-protein level, there's ubiquination, which marks protein molecules for destruction, either because they are misfolded, damaged, or no longer needed. But this doesn't work when those misfolded proteins clump together into plaques. Then you need something like autophagy. But even that has limits, and a cell that lives for decades tends to reach those limits.

As for Aβ possibly being the body's way of flagging problem areas, that seems unlikely, because such a function probably would not involve drastically changing its 3-D structure (i.e. going from "correctly folded" state to "misfolded" state), and certainly would not involve forming plaques. These and many other observed features are better explained as features of the pathology and not just a kind of damage signal reporting on the existence of a problem.