Braxton didn't know her, because he was from an alternate future. The entire timeline up to just before the "Temporal Explosion" (gotta love those threatening plot device) has always been the same. Starling and the Aeon crashing on Earth are part of it.Once Janeway blew up the Aeon, there was no Temporal Explosion and Braxton never got in a fit of rage to destroy Voyager.
I can't speak for the writers, but I see it like this:
When Voyager encounters the Aeon, there has been an explosion in the 20th century caused by Braxton. That means it's already part of normal history. The only difference in the end is that 29th century Earth will survive.
Also, at no point did Janeway talk about restoring the timeline, has if she had acknowledged the fact Starling was part of her history.
However, the catch is, by blowing the Aeon (instead of letting it go, which caused the big boom), Voyager slightly changed its own past. So at the end of the episode, Voyager would normally BE in an alternate timeline. However, I assume here that any change in the past drastically changes the future (the chaos theory). Still following?
But we all know that the writers don't see it like this. Remember "Trials and Tribble-lations" where Sisko and the others interfered, changing the original timeline without any problem. If we hadn't seen that they weren't there before (in "The trouble with Tribbles") it would have a predestination paradox. As for now, I just don't know anymore...
On a side note, I would like to reiterate my hatred of the Federation Temporal Police, the temporal "Reintegration", the ever-self-contradicting paradoxes and the fact that Time Travel is that easy. If I were to have my way, I'd have about 3 or 4 time travel stories in the whole series, and good ones like "Yesterday's Enterprise", "Future's End" (Yes, I did like it, unlike the sequel "Relativity"), "Year of Hell" and the one with the Polaric Ion Explosion.
You'll notice that even though in some of those instances there are unresolved paradoxes, the story is generally good enough to ignore thoses issues. Which isn't the case in "Relativity" in my opinion.
Oh slag, I gotta learn to write shorter posts...
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-If you took that broomstick out of your tail-pipe once in a while, you might have some FUN for a change!
*Rattrap - Beast Wars*
-Let the Fates land where they may!
*Megatron - Beast Machines*