One of the iconic "dungeons" in Dark Souls is Sen's Fortress. It's safe to assume that a being named Sen ruled the fortress, but we do not see Sen or get any dialogue about him/her/it. The snake warriors guarding the fort can also be found throughout The Duke's Archives, which is home to Seath the Scaleless. This leads me to believe that Seath may have had some involvment in the construction or rule of Sen's Fortress. Just looking at the names; Seath and Sen are very similar. Perhaps the snake warriors view Seath as an ascended snake that turned into a dragon, just like in Chinese (or maybe it was Japanese) mythology. So maybe Sen is what Seath's name would have been if he were a snake.
Here's another thing to consider. All non dragons before the first flame sparked, were hollow like. When Nito, The Witch of Izalith and Gwyn found there Lord Souls they became Gods. Hollow (hollow like being) + Lord Soul = God
During the Age of Fire Gwyn split his Lord Soul into nine shards to be given away. Four were given to his knights; Ornstein, Artorias, Gough and Ciaran. Another four were given to the Four Kings of New Londo, while the final was given to Seath. Seath appears to have been given the largest shard of Gwyn's Lord Soul out of the other eight (consider that The Four Kings soul fragment was split into four equal parts for each king, but then fused together after the fight). What I'm getting at is perhaps Seath actually got a regualr sized fragment like the others, but as the first flame ignited Seath may have only been a snake while he found a powerful soul as well (obviously not on par with a Lord Soul). Snake + Powerful Soul = Non-Immortal Dragon?
While there's no way to prove this theory true I'd like to think that this is the reason why Seath doesn't have immortal stone scales, because he's not a true dragon but an evolved snake.