The mouth of Sauron has spoken

[BrianSoddingBoru4]

They can fly high. But then you have to factor in Ringwraiths and Sauron's command of other winged beasts who can also fly high. Also, Sauron's power over the wider world may be diminished but his control over the land of Mordor is potent including its weather so there number of ways he could impair the eagles though in his weakened state it would be taxing. So even high up they are not safe.

He's only got nine Nazgul. And what if the eagles equipped for IFR?

True there are only nine Nazgûl, but we don’t know how many of the airborne fell beasts Sauron had in reserve. It’s like saying the Magnificent Seven only had access to seven horses.

Boru