The same reason the Council at Rivendell didn't just get the giant eagles to fly over Mt Doom and drop the ring in that way.
Because Sauron could disrupt them with arrows and Fellbeasts causing them to drop the ring somewhere in Mordor? Or because the eagles had their own agency and wouldn't be willing to perform such a dangerous mission while Sauron lived?
I'm not sure how either of those would apply.
