Your point was that Sauron can detect the ring at a distance. He can't, or he'd have known that Aragorn didn't have it. The fact that the eagle would be spotted is irrelevant, as shown below.
Yes; "hordes" is an exaggeration. But even nine-plus would be plenty to slow down the Eagles.
Ok, we've made another advancement. We've now slowed the eagles, instead of stopped them.
Because the flying party is easy to spot. Simple. Eagles aren't recon planes; they can't fly above detection level because they need to breathe. A party that entered Mordor with Aragorn's guidance (it's pretty clear that he knows of at least one or two secret ways in) could sneak up to Orodruin without being detected... maybe.
But once more, the issue is whether the Eagles would be stopped *before* they reached Mount Doom.
First, they can't be, period: Sauron doesn't have the means. Ballistae and bows aren't SAMs, they can't hit an eagle flying high enough, and enough would be like 1000 feet. Surely youre not suggesting that at 1000 feet you'll suffering from altitude sickness. Second, it doesnt' matter if they are spotted at all, since it takes relatively very little time to fly from the black gate or whatever Sauron thinks it's Mordor air space to the Orodruin, as it's said in the books; much less in fact than it'd take Saron to figure out what the hell are they doing, send a patrol, and the patrol to arrive at the volcano Sauron didn't have a reason to guard at all.