They could have approached Mt Doom from the side opposite Sauron’s eye. Landed and snuck into the Sammath Naur. Tossed it into the lava and it’s adone deal. Then they would be home in time for mead/cornflakes.
You’re assuming that the Dark Tower was shorter (or at least lower on the lay of the land) than Mount Doom here. Sauron could see past the mountains on all sides of Mordor, remember, due to the tower being located on the volcanic plateau of Gorgoroth. The Eagles still had to fly past the big Orc strongholds in Durthang, Minas Morgul and Cirith Ungol tower, too; and by the Sea of Núrnen there were colonies of enslaved men and lower-ranking orcs that were working Mordor’s only arable land.