You are correct - air bags are not enough and are not intended to be enough. That is why they are called SUPPLEMENTAL restraint systems and that is why the warning on the visor that is there continuously tells you to fasten your seat belts.
Your comment that the airbag should have deployed because you suffered an injury is one that many, if not most, people make when they suffer an injury in a situation where the air bag has not deployed. This stems from a misunderstanding or lack of knowledge of how air bags function.
The steering wheel and passenger dashboard airbags are not designed to deploy in every situation.
Several parameters must be met before they will deploy. There are 3 sensors that sense the rate of deceleration, and at least 2 must detect crash conditions. This prevents air bag deployment in non-crash conditions.
The point of impact has to be within a certain range of a head-on collision. If the vehicle is hit from the side or rear, the front air bags will not deploy because the forces are lateral. Side air bags and side curtain air bags are designed for side impacts.
There is a vehicle speed threshold that must be passed before the air bags deploy. Air bags are not designed to deploy in a 5 MPH accident because a properly belted occupant should easily survive an accident at that low speed without airbag deployment, and such a low threshold would cause many false deployments.
Most people do not realize that air bags provide little or no protection without a properly fastened seat belt, and in fact may cause more severe injury or death without seat belts. The seat belts keep the occupant in a position where the air bag's cushion will do some good. Without seat belts, occupants can be ejected from the vehicle, pushed down into the footwell, or pushed over the steering wheel through the windshield.
It sounds like you were fortunate and did not suffer a life-threatening injury.