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).
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.
The attitude adopted by much of the r.a.b.t community was a deliberate echo of Tolkien's own: we treated Middle-earth as an independent "sub-created" world (to use Tolkien's term from "On Fairy Stories") which ought to have "the inner consistency of reality". In such a vast imagined history, there are obviously huge gaps in our knowledge where Tolkien never told us what happened or why. But in a few rare cases when those gaps are closely connected with the story that Tolkien did tell, we can make a strong case for a unique consistent way to fill the gap (or at least a "very likely" answer).
One of my favorite examples of this was a longrunning collection of debates related to the blades that the hobbits got from the Barrow Downs and their effects on the Nazgul. I won't say we reached consensus, but after years of related discussions quite a few of us had reached the conclusion that those weapons had some sort of enchantment making them especially dangerous and frightening to the Ringwraiths. And then, unexpectedly, a previously unpublished excerpt from one of Tolkien's essays was quoted in Hammond and Scull's Reader's Companion that explicitly confirmed our conclusion (and much of the related evidence that we had gathered). It felt for all the world as if a scientific theory that I had helped develop had just been confirmed by experiment.
It's that sort of thing that encourages me to think that we were one step more rigorous than fan fiction. [And shameless plug: I still maintain the newsgroups' FAQ site. It doesn't say much about the Eagles question, but here's the full story on the barrow blades that I mentioned: http://tolkien.slimy.com/faq/History.html#BarrowBlades ]