Okay I would like to address your concerns one by one as I think we're onto something here and we really should nail it down before pitching it to Amazon (I assume they are still in the script writing phase here, and this is a great chance to correct the blatant plot holes in the original (badly written) story).
Karl wrote:What happens if the ring manages to work its way free from Frodo (which it would be trying to do)?
The ring will be completely secured, hidden in a locked box (and preferably encased in molten metal prior to setting out) and attached to Frodo with several chains for redundancy. If anything we could ask the same question of the actual story - where it's pretty clear Frodo took a lot /less/ precautions than I am suggesting.
Karl wrote:The ring gains power as it approaches the Sammath Naur, and it broke Frodo's will there even while Sauron was distracted, so do you not think it might be even more of a problem to have that effect compounded by the force of Sauron's gaze?
Moggy wrote:Frodo was unable to drop the ring into Mount Doom so it doesn’t matter how he got there or how long the rope was.
Behind the eagle carrying Frodo by rope (lets call this the rope eagle, for brevity) is another Eagle carrying Legolas (he is riding this eagle, a rope is not necessary). Neither this eagle or Legolas have any knowledge about the ring or the true purpose of the mission. Legolas is acting under strict instruction from Gandalf to follow Frodo to Mount Doom and to shoot him so that he falls into the magma. Legolas is an established master archer so this task should be well within his ability. As an archer he also has the great advantage of being able to stay at distance from Frodo - thereby preventing the ring from exerting any influence over him due to proximity, as with the rope eagle.
Moggy wrote:An eagle would have taken it for itself. They are not normal eagles, they are basically the same sort of spirit as Gandalf.
As we've already established, the eagle will not know about the ring nor come within any dangerous distance of the ring.
Moggy wrote:Sauron could sense the ring when it got close to him. As Frodo approached Mount Doom, Sauron could sense it was nearby but thought it was Aragorn that had it and that Aragorn was arrogant enough to attack him before mastering the rings power. Flying/pole vaulting/whatever to Mount Doom would have brought the whole of Mordor down upon whoever was approaching. Sure they could have tried the same trick of sending an army to the gates of Mordor while an eagle flew in, but it was just as quick to send a Hobbit and didn’t carry the risk of a powerful spirit being tempted to keep the ring.
You are correct about the need for a distraction. This is why Gandalf himself will take a third eagle (well, preferably as many eagles as he can muster) and approach Mordor from the opposite direction, pretending himself to have the ring.
Karl wrote:What do you do about the fell-beasts that guard the skies of Mordor?
As the Gandalf-eagle distraction is itself airbourne, the fell-beasts shall all be distracted fighting there, also keeping Sauron's gaze off the rope eagle and legolas eagle.