What fake cover story? Most of the people claiming asylum are doing it for the economic upgrade. Why wouldn't that be true for terrorists that are being recruited?

They will still ask you what your name is, where you're from, what happened to you in your country of origin, how did you get here, etc. They actually do an interview, and you have to be able to provide sufficient details about the persecution that you experienced in order to pass that interview. If you don't convince the DHS officer that you had a credible fear, you either fail out right (deported) or they schedule you for more interviews. And you stay in detention all the while. You have to be able to lie convincingly in a one-on-one interview with a DHS official who has absolutely no reason to think that he'll get in trouble for dinging you, a random Yemeni with no legal representation (as opposed to pretending to be a Saudi businessman).

Except have a fake passport. You know those are scanned, right? And that some government has to issue them, yes?

Sure. But do you genuinely believe that a terrorist organization that's capable of doing real harm to the U.S. can't do something like get a fake or stolen or bought passport?