Just to edit fadhatter's reply a little:
Not all of
Elrond's folk were given that choice, only his children:
To leave
Middle-earth with him and choose life among the Firstborn, or remain in Middle-earth as mortals. They could not choose to stay as
Elves.
A rather intriguing choice, as long as Elrond stayed in Middle-earth, his children could stay as Elves (
Arwen was after all quite old), but the moment Elrond left, they had to choose. And we still don't know what became of his sons.
When it comes to the
Eagles, the reason for not using them are multiple:
- First of all, it would have been a very short book.
- Secondly, overcoming
Sauron was a task that the free
peoples of Middle-earth had to feel that they had done by themselves afterwards. After all, from the
Fourth Age, the dominion of
Men began, and they had to start that dominion with the belief that they were capable of overcoming
evil.
- Third:
Tolkien once stated that the Eagles should not be used too often. They were an instrument of
Manwë, and meant to be his messengers and his eyes and ears in Middle-earth. Not his fighting instruments.