I agree that the use of eagles is an unanswered question, that might, to some, seem somewhat surprising that it was never mentioned.
My answer is that Tolkien made the artistic choice to speak of the powers of both Sauron and the Ring in poetic terms only. The last thing he wanted to do is delve into details about Sauron's Anti-craft Aircraft potential when shooting at eagles from throne of Barad-dur, even in a passing comment. He worked very hard to shield our eyes from such details.
Fan fiction is creative writing, more or less by definition: inventing new details is fine unless they conflict with the original story (or even if they do). The author of this article is very explicitly trying to avoid "making up" an answer: he's searching for evidence only within the original story that could constrain the answer to a question relevant to that story. They're both games for fans to play, but at least to my eye the two are worlds apart.
Using other possibilities to refute a possibility goes against the spirit of the article, where the author looks for concrete facts that would render the plan implausible.
In the end, there's nothing I know of (or that the author could find) written in the books that disproves it as a possibility. It simply isn't addressed directly in the text, even where there was a prime opportunity to strike it down (Council of Elrond).