I got the impression from reading the books that even Gandalf and Aragorn had no idea how they could get to Mount Doom, and that there was no way all nine of the Fellowship could have made it without being too obvious and without everyone being overcome by the temptation of the ring. The power of the ring grew as it approached the place where it was made, and I think the temptation of it would have been too much for anyone but a hobbit. I think Gandalf and Aragorn were just going on blind faith, not knowing what was going to happen when they got close to Mordor.
According to JRRT, the eagles were spirits summoned from outside Arda to inhabit the bodies of our flying friends. I always assumed they must have been minor-order Ainur who left Iluvatar’s timeless halls.
And of course Sauron was quite involved! He was a rebel from early on!
I’m not convinced Shelob would have been so easy, even for Aragorn, Gandalf, and company. If Gandalf had let loose with any power at all it would have been a giant neon sign which would have assuredly led to their capture. Shelob was described as having faced down many warriors in her day, none of whom had seriously injured her. The only reason Sam scored much of a hit in her at all was because he was small enough to be beneath her when her own bulk and strength forced her down onto Sting.
I would have to vote for stealth and the end run around the south.
Gandalf hid it in the one place he knew he could hide somethin’.
HIS ASS.
However long their quest took, he wore that One Ring up his ass.
Then before he “died” in the mines of Moria, he gave Frodo the watch. And Frodo hid that uncomfortable hunk of metal up his ass for even longer!