Okay, I’ll let you in on my thoughts.
The artificial world of Little Busters, like you said, transcends time in a sense. This was demonstrated in Episode: Kyousuke of the Visual Novel, where time kept repeating in the real world as it repeated in the artificial world. (I think that’s how it went?) Anyway, the point is that the artificial world doesn’t follow the same ‘time’ as the real world.
Saya, or Aya as she’s really called ‘died’ as a child. But when I say die, I mean she didn’t REALLY die, but experienced something like a near-death experience. Her spirit left her body and entered the artificial world. Why? Two theories. 1: The location was nearby to the site of the bus crash in the future. Just looking at the BG used in the VN and the scenery used in the Anime, this seems to strongly suggest this. 2. Her bond with Riki. An emotional closeness that drew her to the world Riki existed in. As for why it happened even though the bus crash happened in the future? Key magic, the world transcends time, all of that. Follow me so far?
So Aya ends up in the looping world, donning the appearance of Kyousuke’s favourite manga heroine, and entering into his game. The only way out is to kill herself, so she shoots herself, and leaves the world. The time machine is a metaphor created by Kyousuke to explain that Saya got her wish granted. Remember, Aya never died, she was merely in an out of body experience. When she leaves the world, she wakes up from the dream, and appears in bed, rescued by her father. This isn’t a parallel universe or anything, it’s the same world. What we’re seeing is Aya’s consciousness went from point A in time to point B, the artificial world the future Little Busters existed in, and back to point A again. So it’s SORT OF time travel, but more accurate to say world travel, because she didn’t actually enter the future real world, just an artificial world that was created by the future little busters. It’s hard to explain in words, but I hope you understand what I’m getting at here.
She wakes up, and Riki is there. My interpretation is that this is all things that really happened. For some reason, most likely the death of Riki’s parents, he loses contact with Aya and forgets all about her (Typical Key Protagonist Amnesia). As for the CG of Aya with the Little Busters boys in the VN, or the scene in the Anime where Riki meets her in the grass? You could interpret that as Riki finding Aya at some point AFTER the accident, and they become friends again (or lovers) and Riki introduces her to the other Little Busters. It’s purely speculation, but this is my take on it.
Does that make sense?