I enjoyed the thought and analysis that went into this post. It might not be a satisfying answer, but the conclusion seems to be that Tolkien didn't think about eagles as a likely course of action. And if he did, he didn't bother to address it.

It's funny to think that we'd never have this conversation if there had been a line like, "for Sauron can sense the great power of the eagles as they approach Mordor." The fact that fans long for "canonical" answers when there are none is a testament to great writing.