They could have fitted the whole ring / tape / mouse assembly into a small paper bag Aragorn could have kept it in his jacket and fed it little bits of lembas on the way how lovely x
The answer is that the eagles are theologically angels in LotR and both sapient and very susceptible to the Ring, and having a flying Sauron-replacement is not an improvement.
7 points
I just assumed that sauron had some means of stopping eagles flying to mordor
