2001 Subaru Legacy replacing the driver side seat belt
Newer seatbelts may measure how far extended they are or how much tension is on the belt and based on this information, the airbag may or may not deploy.
For example if someone is leaning forward at the time of impact, like someone with her face in the mirror to put on lip stick or whatever, and the person is right up against the air bag... Then the seat belt would "sense" that it was fully extended. And it would not be a good idea for the air bag to deploy in that situation.
If the seatbelt sensor is not connected (or non-sensor belt installed), the airbag might not deploy.
At a minimum, I would replace the seat belt with the same manufacturer's part number belt from a vehicle in a wrecking yard. Might even be less expensive and matching colors too!
Or order a new belt from the dealer. This could cost $500, but I would have no problem paying that. I think my life is worth $500!
If you do install a non factory seat belt, I would post a sign in the car that the air bag system is not working and ride at your own risk. And especially if you ever sell the vehicle, people should know the safety system is not working properly.