Athens relied on grain shipments from the Black Sea because they knew they had no chance against the Peloponnesians in pitched battle, nor could they even successfully attack the Spartan camp at Dekelia. Once Persia is in the war, they can simply occupy the Eastern side of the Hellespont with an army and support it with a fleet; right off the bat, their reconquest of Ionia would deny Athens much of the Delian league revenue they relied on to keep in the war. Even if they didn't commit any forces of their own, the sheer amount of money they could provide their Greek allies meant that as in OTL they could continually rebound from one defeat after another.
But they did have a chance in pitched battles against Sparta, they were all hoplites with the same equipment and the same overall tactics. They could do nothing against the Spartans after 413 because the expedition to Sicily aignificantly crippled the Athenians, before then though it would have taken just some more capable land generals to make the difference.
The Persians, alongside the Spartans, did hold one side of Hellespont, nonetheless Thrasibulos and Alkibiades smashed through them repeatedly, from 410 to 407, when Athens’ forces were much reduced compared to a couple decades ago. And assuning Athens manages to hold onto Amphipolis, she’d have a surplus of revenue from the mines and be rich enough to buy the grain, if worse comes to worse and they can’t dislodge the Persians, although I think they could do so without that much effort under better circumstances.