Way I read it Aeson, at the very minimum retains the small part that makes up his own soul, also the MC was never meant to get the full essence, because Aeson (possibly Saint as well) and the Gods keep hold of the small slivers that are the basis of their souls/powers.
Not that getting the essence will be much cause for celebration since in-game our mc's can already barely handle the teeny, tiny sliver they've got right now that forms their own "soul".
Eh, most demigods don't even have flashy power abilities, I believe they're just naturally tougher, stronger, smarter and more healthy, but for most it kind of ends there.
As for the mc, I think our "demigod" core/powers (whatever those might be) were ironically used to power and/or sustain that blocker spell around our demigod "core". Besides after that coma all our "reawakened" demigod core seems to do is interfere with our already feeble control of our "normal" essence powers, so it seems to be more of a curse then a blessing at the moment.
To answer your question however, had the mc not been under the blocker spell they likely would have manifested some weak powers during childhood, something useless and silly like animating the hand of dead person at a funeral to wave at the crowd or something. It probably wouldn't have been all that impressive, our mc's real powers lie in their "normal" essence abilities anyway, if only we had even a modicum of control and finesse over them.