So I've been thinking about this for a while now. This has probably already been talked about, but I wanted to bring it back up anyway.
Doctor, tell me, what makes the Brigadier so special that there can be two of them in the TARDIS at once. Or was it that the TARDIS could have taken both Amys for at least a short trip, and you just didn't want to. There have been larger paradoxes on board before. Doctor, were you lying? Yeah, you were definitely lying.
I imagine that maybe the Doctor and the TARDIS would have had some paradox related illnesses, but they both have dealt with worse paradoxes before...so you would think they would cope with it if only to help a friend. They did not even have to keep her on as a companion, just let her off literally anywhere else.
But I personally think the Doctor simply didn't want to take her. Older Amy was much more jaded than the younger Amy was. She wasn't as ideal of a "companion" for the Doctor to have anymore. It was much easier to take younger Amy away under the assumption it would make it so older Amy never existed...with an interesting intersection of Time Lord morality and the Doctor's own selfishness.
Because as we know that the story changes but the ending remains the same, at least for the CIA. The Doctor has stayed in line with this motto sometimes but has broken it for others...but I digress.
If the ending had to remain the same, then it should have been younger Amy that the Doctor left behind, so she could later become old Amy. They saw that she was part of "the consequences of an event" and thus shouldn't have been able to influence it any longer. But as I said before, we need to incorporate back in the Doctor's selfishness and his desire to have a less jaded companion.
So, to recap. The Doctor didn't want to take both Amys in the TARDIS because it would create a big paradox...meanwhile, he is helping a different paradox to persist by taking younger Amy and hoping it erased that timeline. Which by Time Lord standards was the incorrect thing to do! Keep in mind Amy and Rory's timelines are pretty complex.
After all, they saw themselves waving at them in 2020. This would seem to imply that the Doctor may have known that older Amy wasn't the end of the story.
But of course, this is coming from the Time Lord that purposefully tears the last page out of every story they read. The Doctor hates endings.
What is also possible is that had the Doctor taken older Amy, she may have eventually become that Amy they all saw waving with Rory. Even if age was a factor, we have seen time and time again that physical appearance, including aging, can be changed through a variety of circumstances (from instances like Richard Lazarus to Nyssa of Traken).
But we don't know if this would have happened for certain. What we do know is that Amy and Rory died before 2020 anyway, so how did they wave at each other? Were they still traveling with the Doctor on some unseen adventure? Or did this get erased when Rory did when he later fell into the crack? Uncertain, all we know is that their timeline is so complicated.
The Doctor had already taken part in changing all sorts of story endings before (which is perhaps why he doesn't like to know them...plausible deniability). But we do know that what would have been considered the "correct" thing to do by Time Lord standards would have been to leave younger Amy behind. However, since the Doctor has his own agenda, he wanted to take the version that still had oodles of faith in him. He developed that lie—because the Doctor lies—to excuse the fact that he did not want to take both Amys on board, even though we have seen that two Brigadiers can coexist at least temporarily on the TARDIS, and both the Doctor and the TARDIS have weathered worse paradoxes for friends. Maybe he was trying to make the paradox he was creating by taking the younger instead of the older Amy with him less severe by trying to erase her rather than having a fragment of a now-aberrant timeline running about. It simply wasn't as convenient for him to keep the version that did not really have any faith in him whatsoever.
Hmmm, I wonder if that aspect of faith comes back up....*looks at the episode that came right after The Girl Who Waited*
I threw this short essay together in about 15 minutes, so don't come for my throat if I've missed anything please haha