Both the pro- and anti-eagle arguments make assumptions. The pro-eagle argument is built on assumptions about the limits of power.

We know that Sauron is unequaled in power among the Maia. We know that Sauron built mountain ranges. We know that Saruman easily overcame Gandalf. We know that Gandalf has powers that allowed him to overcome a Balrog.

Tolkien has made the artistic choice to shield our eyes from overt magical power. It is the nature Middle Earth that great magical power is a very real thing. The lack of clear evidence of the exact limits of magic is because Tolkien really did not want to discuss it, and that is not a good argument for simplistic assumptions.

Assuming the non-existence of great magical power in the hands of Sauron is just foolishness. It boils down to whether a guess about the limits of Sauron's power is more believeable than a guess about the potential extent.

I would further note that the Council of Elrond may have no firm knowledge either. In the end, they guess that secrecy is the best hope.