What is that statue hypothesis based on, really?
It's pretty clear in the prophecy that the White-Gold is kaput. If the statue had the same properties as the amulet, the tower would be very much intact.
Which prophecy?
I'm basing my statue hypothesis on this:
1. The motivation for most of the actions of the Champion of Cyrodiil was that the barrier to Oblivion was weakening, and needed to be restored.
2. To restore the barrier, it was necessary for the newly crowned emperor, in possession of the Amulet of Kings, to perform a ritual to relight the Dragonfires at the Temple of the One.
3. Martin Septim, just recognized as emperor, had the Amulet of Kings, and reached the Temple of the One, but was unable to perform the ritual. Instead, he did something with the Amulet, which seems to have resulted in his transformation into the avatar of Akatosh. In other words, all the steps were performed, except for the last. Instead of ending up with a living emperor, wearing the Amulet of Kings, and a burning Dragonfire, we ended up with a statue of Akatosh.
4. The Daedric invasion ceased, the Oblivion Crisis ended, and we are told, the barrier to Oblivion is restored. (Some characters in Skyrim claim it's stronger than ever.)
5. So, in sum, we have one thing, the statue of Akatosh, in place of other things, the emperor, the Amulet of Kings, and the Dragonfires, and apparently performing the same function: maintaining the barrier to Oblivion. That leads me to conclude that the statue is equivalent to emperor, Amulet, and Dragonfires, and therefore functions as the Stone of White-Gold Tower.