Concerning Tom Bombadil (who frankly I always loved), the main issue is that it is unclear what he is supposed to be, beyond who he is. If Tom Bombadil is the Green Man, a stand-in for old goodness in Pagan Faith; nature by the view of man (or hobbit kind); giving him the ring wouldn't work because he would not comprehend the situation. Not because he is foolish, but because he is so alien to the concept of it; and of course if he was "corrupted" by it things would get infinitely worse. Tom Bombadil is something that cannot be matched, namely because he's larger than life by way of viewing from people who are far quainter than anyone's life is. And while he is not corrupted by the ring, he is in and of itself his own entity who could be corrupted by the promise of things to come. Which fits the theme of the story.
Gondor wants the ring, as a weapon; but it corrupts Boromir who uses his last bit of humanity and free will to see those predetermined by fate to survive. Galadriel considers it, but she'd be corrupted by it and realizes this. Gandalf is offered it but he knows he could be corrupted by it and is terrified of that fate. Tom Bombadil does not want the ring, but he could do the deed; but he could also listen to Sauron and say in a flight of fantasy "All right, evil sounds like the order of the day for me too."
A lot of people talk about it, especially if you get into Tolkien scholarship properly. Tom is a very polarizing figure. I'd still say the Eagles aren't a -huge- problem, just because the real toll of the war on Frodo and Sam doesn't happen until the Scouring proper. Gollum is, to me, less deus ex machina, and more a traditional protagonist without being given the title. He drives the movement forward, and it is clear he is (like the rest of the Fellowship sans Boromir) fated for something. There's a lot of Catholic themes brought up with Gollum, namely commensality, and it is made clear rather early by way of Frodo that they're supposed to be mirrors. Gollum will be redeemed through his corruptions and die, whereas Frodo will become corrupted through constant sacrifices, and yet live. Which is why Frodo is really a very tragic character once the end of the stories happen. He's so burnt out.
But it has been awhile since I wrote those papers. Mostly been grumbling about dwarves for the past few years. I don't disagree with you on how they can be seen; but intent is important and with a broader reading it does make sense within Tolkien's works. There's an internal cohesion to it.