That theory is B.S. The ring contained Saurons power, and also gave him power over the Nazgul (former kings who he gained power over by using other rings). The Nazguls had incredible power to detect the ring and would have easily noticed the eagles flying straight in. Once the ring was destroyed, it broke Sauron and the Nazguls power, so it was easier to fly in pick up the hobbits and get out.