- He probably managed to recover, but his shit luck kept leading him to getting hospitalized over and over again.
Thus, Homunculus is trapped in a time loop where he is created, sealed up, lost in a cafÃ©, given back to Eike, given to Dr. Wagner, created, sealed up, and so on. However, Dr. Wagner's soul is still bound to Homunculus. So when Eike finally dies after the game's events (as in Ending C, or some future occurrence), he's brought to Homunculus. However, Homunculus only exists in a time loop, so this future Eike must be brought to a point in this time loop. In this case, that's the start of the game.
This is where the Extra Prologue starts. This Eike is from after the end of the game, so he remembers everything that happened in the game. This allows him to take the proper steps to either stop Homunculus from being created (telling Wagner to make the elixir) or removing him from time altogether (throwing the stone at him).
Sure, it's not perfect (I still haven't figured out when Homunculus has the time to be out and about and switching babies), and it wouldn't explain theoretical future Eike replacing original game Eike during the Extra Prologue, but I think this is pretty close to a "real" ending.
- Makes sense. I don't know where people keep getting the idea that the Wagner thing is a seperate/alternate timeline from (unless it was Word of God). Personally I've always seen each ending as possible, given Homunculus explained that for every action in timeline, another, different action occurs depending on the choices made. So in one timeline, Eike fixed everything (returned the right girsl to their times, stopped his murderer, and defeated homunculus), while in others, he died, got piss drunk, doesn't solve anything and only prevents his death.