I guess there is a question of the nature of its incarnation. Some say the Fossil itself is being animated into a Pokémon, its stony shell loosened from the rock it's embedded in and serving as a shell for a newborn creature. Others again believe that DNA is simply extracted from the Fossil, and then used to create an entirely new Pokémon (or perhaps reanimate it the way it was before it died and became fossilized).
I stick with the latter theory. The Fossil itself remains intact, and has been handed over to its final Guardians, the fossil researchers who will store it, praise it and consult it for eternity, long after the Twitch stream is turned off. It guided Red from Mt. Moon to Cinnabar, through victories and hardships, and has now been given an eternal slumber place on Cinnabar Island. From its depths the Guardians recovered enough information to create an Omanyte, a deity become flesh, Helix incarnate, a helpful companion to follow Red through the final portions of his quest.
By the way, I kinda wish for this stream to beat the Champion, roll the end credits... and then start up Pokémon Gold/Silver. At the end of that game, Red can be battled. It would be a trivial matter to edit his team to match the one of Twitch Red at the end of this game.