But the Enterprise C needed an experienced tactical officer otherwise they would have been destroyed by the Romulans all too soon and failed in their mission to protect the Klingon colony and prevent a Klingon/Federation war. There were no experienced tactical officers onboard the Enterprise C and Tasha was willing to take on that role if it meant saving billions of lives.
That's why Picard let her go, not really why she wanted to go, (To have a more meaningful death) & certainly had nothing to do with why Guinan told her about her alt-universe status, (As well as influencing her attitude by telling her that her death had been meaningless) because Guinan wasn't privy to those detailed events of the mission when she blabbed. I interpret her reasoning as simply being that the situation bugged her, & when confronted about it by Tasha, she spilled the beans, because she let the truth from her insight get the better of her, which is in no way related to her involvement in the mission, by informing Picard, to try to restore things properly, & the end result is that things never actually do get restored completely to how they had been, & for the first time ever in Star Trek, we carry forward in an altered time line, with the birth of a time traveling Tasha's daughter existing in their realm
I always got the sense when watching 'Yesterdays Enterprise' that Guinan was thrown off balance by the presence of Tasha after all she has never met Tasha and knows that Tasha should be dead but at the same time has memories of being good friends with her and is having to deal with the emotional consequences of that throughout the episode. When she first encounters Tasha she looks positively freaked out (by Guinan standards) and when they later talk Guinan is more driven by emotion than reason.
Emotional discombobulation is a fair point, but in Time's Arrow, her own life is on the line, reason enough to be emotionally invested, & yet she purposely refrains from involving herself in the events, for fear of negatively affecting the time line, which is exactly what she does in Yesterday's Enterprise, when she involves herself by offering Tasha info that she shouldn't know, & actually didn't know, or wouldn't have surmised had it not been for Guinan's odd behavior & an off hand remark. When questioned about it, she could have just as easily treated Tasha as she did Riker
The real interesting thing is that her new philosophy in Time's Arrow shows up after she learns about the daughter of Tasha Yar, which she points out to Picard in Redemption, & indicates that she knows that they were involved in the turn of events. It would be easy to think that she feels responsible for Tasha's alt-universe fate, & chooses to not make similar mistakes from then on