The remaining forces that were left after the battle at Pellenor Fields marched towards the Black Gate knowing they couldn’t win the battle. They did it because they knew that Frodo and Sam couldn’t make it to Mount Doom without drawing out Sauron’s army and at the same time it kept him blind to all else that moves.…

Here another reason why the eagles didn’t take frodo

The biggest point everyone has missed which was stated in The Hobbit An Unexpected Journey