I don`t think Steve had problem with BW`s past, he actually trusted her, they even had internal jokes and small talk while working, but when he found her doing a mission of her own, that she never told him about, he realized she was like the rest of SHIELD, which is why he goes on all alone until he teams up back again with BW and she proves to him he can trust her, but he never mentioned having problem with her past, he did question her methods of not telling people the truth, even those that are supposedly close to her, which is why he told her it is hard to trust someone who hides things all the time.
I don`t think he cares about people`s past, in his conversation with Fury he admitted they did stuff in WWII; I think he cares about people`s intentions behind the doing of certain things and who they are now.
But there are two versions of Cap in the MCU, the one by the CA movie writers and Whedon`s Cap in Avengers; I mean Whedon`s Cap is not too judging of some, at first meet but, he was very much of Tony because he knew his father in ways that not even Tony himself, knew of his own dad, so he was very judging with him...
I hope they fix this in MCU, I personally prefer Steve as he is written in the MCU, the writers get the character`s heart and respect Evans` inputs too, while Whedon, so far has been odd with the character, I hope that changes with AoU.