Oh, did Avatar end like that? Or you mean the fact that its known there is this special power which is transmitted?
I get the not revealing the secret identity thing, but what's the problem of telling the power some hero is using is the same All Might was using years ago? That doesn't ruin the hero's secret identity at all. Leaving aside the fact that heros in BnHA don't seem to care about secret identities except for All Might.
Not the power itself but the fact that it can be given away. If some decent villains knew that it wouldn't take long for them to add 1+1 for All Might (or the previous user of OfA) weakening and/or vanishing with new wonderkid in town appearing. when you connect the dots you can pretty much mess around how you see fit.