Even a parallel construction has limited uses, since you can't use the same excuse every time. The NSA probably doesn't trust the FBI to come up with something plausible.

That depends on how apathetic people are. Most of the time they will accept a completely garbage story and you can create infinitely many of those.

Sure, and that’s why juries buy the same line from cops every time, but major actors actually affected by potential backdoors are likely to be a bit more skeptical and start digging around for the real reasons.