I too was very skeptical about why the doctor needed to involve all of his past regenerations in order to save Gallifrey.
When the 11th doctor decides change his own past and save his home planet, the dialogue is as followed:
The War Doctor: "There's still a billion billion Daleks up there
11th Doctor: "Yes, there is! There is!"
10th Doctor: "But there's something those billion billion Daleks don't
11th Doctor: "'Cause if they did they'd probably send for
Clara: "What? What don't they know? "
11th Doctor: "This time there's three of us."
He gathers all of his past regenerations in an effort to freeze Gallifrey as seen in the following moment:
General: "It's delusional! The—the calculations alone would take...
hundreds of years!"
11th Doctor: "Oh, hundreds and hun—"
10th Doctor: "—dreds, but don't worry. I started a very long time ago."
It doesn't make any sense that this calculation can only be running for the duration of the doctor's life. Couldn't the 11 doctor go back to say the Jurassic Period, burry a calculation in the ground, and then travel to the future and get the result?
And let's say the doctor needed the TARDIS at different points around Gallifrey in order to do the freeze, it still doesn't make sense why he couldn't have just kept revisiting the same point himself, and have a ton of Matt Smith doctors solving the problem.
After giving it some thought however, here is the only valid explanation I can come up with(other than the obvious: "It was cool to have all the doctors"):
The doctor cared about having all of his faces involved in the saving of his home planet. He wants them all to be "The Doctor".