First of all, you have to understand how airbags work.
Airbags are SUPPLEMENTAL restraints, the seatbelts are the primary restraints. See the "SRS" on the bag covers? That stands for "Supplemental Restraint System". The seatbelt locks are the same thing.
Airbags are very violent things and can cause a great deal of injury themselves. The systems are designed to be used only when the crash forces are such that they will help more than they hurt. It is entirely possible, and likely that you could be in an accident, have no airbags deploy, and have the system operating properly.
By the virtue of the fact that you are here and alive, I bet the system operated fine and the crash forces were such that deployment of the bags and seatbelt tensioners wasn't necessary.