Regardless of any law, the outbound scan/inspection of documents is for the airline to confirm that you will be able to enter the country you are arriving at; they do not want the fines, inconvenience etc of having you refused entry because you did not have the right documentation.
Anyway, I agree with the others, try to get the passport before you leave since if anything goes wrong in the process you'll be stuck unable to return as you will have no proof to the airline that you may return to the country. Since you will have no greencard you would have to use a visa(-waiver) and Immigration is sure to spot this when you arrive. As stated by others, if you are a citizen you must use your US passport to enter; not having one will be trouble.
Speaking of which, I used my US passport for the first time last week for a trip home to England. Of course I used my UK passport for entry to the UK, but I showed my US passport at all other times. After 11 years of coming in/out with an H1B, GC etc, this was without doubt the easiest and most pleasant ever entry to the US - well worth all the hassle of waiting.
Paul