My wife, who is a permanent resident, did need to show her passport when she came back to the US, even with a green card. The CBP agent looked at her card first, then checked (and stamped, even though he wasn't supposed to, according to the other agent present) her passport.

I'm sorry the NICOP application is offensive to you (for the record, I wouldn't answer that question, either). It looks like it's definitely going to limit your travel prospects..