Once we have a parsing table, we can parse (or reject) any sentence without any reference to the grammar. So at that point, the fact that the grammar was or was not augmented is essentially moot. (It would be meaningful if there were a user semantic action attached to the augmented start symbol's sole production, but that seems impossible, since the augmented start symbol's production was added automatically, not by the user.)
And it is indeed the case that most parser generators do in fact optimise their parse table by making the shift of the end-of-input marker the accept action, rather than waiting for the augmented start symbol's production to be reduced. With that optimisation, the augmented start symbol is never used in a parser action, so the symbol itself need not exist. If the parser generator augmented the grammar, that augmentation has essentially been undone, except for one little mystery: what is this end-of-input symbol which can be shifted? It doesn't appear in any reducible right-hand side.
Anyway, the point is that it's not parsing which requires an augmented grammar; the augmented grammar is necessary to create the parsing table. The cases where it is necessary are essentially the cases in which there is some non-default reduction action associated with an end-of-input symbol lookahead. That reduction action could only have been correctly added to the parsing table through the analysis of a state which includes the augmented start symbol's production.
(Strictly speaking, as alluded to earlier, the end-of-input symbol can't really exist in the parsing table unless it is present in some right-hand-side in the grammar, and it isn't present until the grammar is augmented; the augmentation not only adds an additional non-terminal, it also adds the end-of-input symbol itself.)