That's not right either. Thorondor was sent by Manwe to watch over the Noldor but there's no place in the book they've been classified as Maia. Gandalf wasn't a Maia either, he was Istari. Thorondor's classification has always been Eagles of Valinor.
Gandalf doesn't gain the right to fight Sauron directly by becoming white. Since Saruman had fallen, he comes back as the leader of his order, like a promotion. Istari were sent to earth in the aid of Elves/Men to defeat evil. There are no guidelines as to how to fight them. Each of them fight in their own way, Saruman studied lore because his power was knowledge. Saruman should have got Narya, but Cirdan the shipwright gives the ring to Gandalf because he perceives Gandalf is the good guy. Narya's power is to inspire others to fight tyranny, that's why Gandalf is always motivating men, elves, dwarves and hobbits. Him being Istari and a wizard, he is already powerful, but with the ring too, he's able to do it even more effectively.
The eagles plot could well be true, but that's a fan theory and that's all. Eagles only come to Gandalf's assistance in grave danger and they are a proud race of Valinor. They would not bear load as if they are common mules. They also do not partake in any schemes by Men willingly because it's not their job.