[Mithadan]

This is a frequently discussed question. The obvious answer, that it wouldn't be much of a story then, doesn't serve here. Otherwise, there are several possible responses.

1. The eagles were not under Gandalf's control. In the Hobbit, they were attracted to the fir wood by the fire and commotion, and the Battle of Five Armies by the din and hue of war. Gandalf is rescued from Orthanc by an eagle who journeyed there at Radagast's request and later from the top of Zirak-zigil perhaps at Manwe's request.

2. Assuming that Gandalf did not know of the airborne Nazgul, Sauron did nonetheless have spies in the air in the form of crows, etc. The Ringwraiths were abroad and were capable of sensing the Ring. A flight of eagles bearing Frodo and the Ring would eventually be detected before reaching Mordor. The eagles could not fly the whole way at once so there was danger of assault during a rest stop and again when the eagles reached Mordor.

3. Gandalf knew of the Nazgul, or suspected that Mordor was otherwise protected from intrusion via the air. A flight of eagles would have been very open and obvious as it approached Mordor.

4. The eagles could not carry such a burden over such a distance or they would not do so. Also, the eagles were only nominal allies; the Hobbit notes that some are not trustworthy.

I'm sure I'm missing a few possible reasons. Welcome to the Barrow-Downs Ulmo!