I have no problem with Hurt, Tennant and Smith's Doctors realizing they needed more computational power to create a planetary-wide stasis field. So they went back and convinced past doctors to do the computation and to show up.
I know some are worried about past doctors knowing the future and the consequences of it. But I feel it shouldn't be too hard for the 3 doctors to convince the past 8 doctors that what they are doing is absolutely critical to Gallifrey's survival and to do it with the TARDIS on autopilot and sensors completely turned off so they don't know exactly what's happening or when it happened.
Capaldi's doctor shows up because he already knows he's needed.
I had a different read on it, based on the fanwanky trailer.
Matt Smith says that there's something the Doctor has been running from "all my lives." He can't be speaking of "The Moment," because he's only been running from that for three lives.
What if he's always known, even if only in a subconscious way, that he'll be there when Gallifrey falls? That he's been running all of his lives from his future, not his past?
And, of course, Capaldi's there because he needs to be there. Perhaps the calculation doesn't even finish until Capaldi's life.