I think the answer is rather straightforward and is hinted in other encounters - eagles are getting involved as a last resort, to maintain the balance of good and evil. They are seen as the open messengers of the divine power, and aren't getting involved when the balance can be maintained by other means. So since there was a way to bring the ring to Mordor (even though an extremely hard one), eagles didn't interfere.
I.e. it can be seen as a way to preserve the potential for choice between good and evil. Open and obvious involvement of the divine diminishes such potential, so it happens only when really necessary.
Interesting theory, however, is there any example where the eagles come to the last minute rescue of Sauron's forces in order to maintain the balance of power?
Eagles are seen only as messengers of good. In general that would be a question only from the perspective of dualism, but I don't think this is the case with the Tolkien's view on his world. The world appears predominantly evil, but in essence Tolkien maintains the view of the ultimate good (i.e. hidden), as an intention of Eru. So tipping the balance into the good side usually doesn't disrupt it as much. However extreme shifts still can have serious consequences. Valars getting involved in banishing Morgoth destroyed Beleriand altogether. The fall of Sauron diminished the Elven presence and so on.
I.e. it can be seen as a way to preserve the potential for choice between good and evil. Open and obvious involvement of the divine diminishes such potential, so it happens only when really necessary.