I don't think you are insane, I think the writing intentionally echoed their first scene together, or rather that it was deliberately ambiguous when Echo looking at the screen said "she's my friend. Friends help each other." I'd rather have had Echo not say that line actually and yet help Sierra. The echo would have been less obvious, and thus more subtle, but I suppose that making Echo repeat lines like that suits her name and gives her that robot-like talk recalling that her behaviour has been programmed.

Also perhaps those little lines that she keeps repeating represent a bridge between her blank state and her doll personas, words that break out of the Dollhouse, words that might de-trigger her eventually, freeing her, unlike the word "treatment" that makes submit.

As for her not following the mission's objective, well, in a way she did help Rayna too whom she was programmed to help, as Madame Boss (can't remember her name) pointed out in the end, soyt he line worked for both Sierra and Rayna. She killed two birds with one stone.

Thanks for dropping by and leaving a comment by the way!