"The Eagles are a dangerous 'machine'. I have used them sparingly, and that is the absolute limit of their credibility or usefulness. "
– Letter 210, J.R.R. Tolkien
As they were principally messengers of Manwe, some kind of spirit in eagle form, they may have had the same limitation that Gandalf would have had re interaction, though they certainly did intercede on occasion. The Eagle who bore it could have become corrupted etc.
So, in one sense it appears to be an obvious solution, but it may not have been a sound choice.