Please excuse the stupidity in the question if there is any. Why can’t we use the average acceleration formula for centripetal acceleration? Why do we need to derive another formula for it. In its derivation (in school) we even equated it to the average acceleration formula.
Because here you assume that the object doesn't start and end at the same point. Say I were to start and end at the same point on the circular path. Then $\Delta v$ = 0, as the magnitude and direction are the same, and therefore the average acceleration would also be 0. But that cannot be the case as the velocity is continuously changing (in its direction in this case). Don't forget that velocity is a vector quantity, meaning that it has magnitude/length, and direction. The change in velocity is towards the center, as the object is being "pulled" to the center which is why it continues in a circular motion. This is why you must take into account that this is in fact not a linear displacement but a motion of rotation.
Hope this helps!