Actually it's Alfred who leaves us with that impression, and if you can't trust Alfred, who can you trust?
Of course, I'm biased; I think it's nonsensical, OOC for the boys, and insulting to everyone involved that they wouldn't tell Babs who she was working with or who this guy who knew their secrets was (actually I'm always a little hazy on whether Tim knew anything about Babs pre-uniform - her identity doesn't follow immediately from Dick's the way Bruce and Jason's do, and he'd have no reason to assume the retired Batgirl had any current relevance to the team anyway). The longer that continues the more insulting it is, so the earlier Babs figures it out the happier it makes me.
though comfortable with the kid.
I imagine it was a huge relief to her to finally have someone as nerdy as her on the team. They're such cute dorks together, they never get enough shared panel time.