The Great Eagle Mystery

I think you are ignoring the obvious fact of the ring itself. We know the ring has different levels of influence over different species. The reason Gandalf didnt take it apon himself to carry it was that he would not be able to resist its power.
I dont remember any direct evidence of this with the eagles, but its certainly possible that they have a low resistance to the ring's power, so they could not be trusted even with carrying one who held it.