Actually... you do need a passport to depart the USA by air. The airline has to swipe the passport to compare your name/identity against various government watch lists (for outstanding warrants, suspicious activity, unpaid bills or child support, etc). If you are okay to travel, your boarding passes will say "DOCS OK" and you can go.

But that's a separate issue.

The main issue here is that the airline didn't want to board the OP because if he/she had arrived without proper documentation at the host country and got denied admission, the airline would have to pay to fly him/her home (because it was the airline's fault for letting him/her get on board without confirming eligibility to enter the host country). The airline doesn't want that kind of financial liability.

That's why the ticket agents always check your passport/visas/etc to make sure that you have no document-related reason for not being admissible to the host country.

Of course, if you get denied entry for another reason--because you violated some local law, or were caught with illegal items--then it would be YOUR fault (not the airline's), and YOU would be liable for the expenses associated with your repatriation to your home country.