It sounds more than a bit unlikely that Starfleet would lack
an edge over the Klingons if it has enough ships to spare that one of them would "stumble" onto Khan...
And how would that ship know that Khan was worth anything? The Botany Bay
is just a run-of-the-mill derelict; even if ships of that type weren't supposed to be capable of transporting people from star to star, TOS showed such a ship in interstellar space hauling ore in "The Ultimate Computer", suggesting most skippers wouldn't blink at the sight of one.
It might be a bit more plausible that S31 would send a dozen Fox Mulders into the archives, looking for "paranormal" gamechangers, and one of them would patch together the story of Khan plus a few educated guesses of where his mythical ship might be found...
But olSpock could save them that trouble, as he seemed quite interested in giving a personal edge to nuKirk, and warning him about all the early 2260s threats would be a prudent way of doing that.