FYI...If the system is in a faulted state the RCM (Restraints Controller Module) with any pyro device (air bag, pretensioner) the RCM will still attempt to deploy the device faulted or not. An unplugged/faulted seat device(Seats contain a belt pretensioner on the buckle side and optionally a side air bag)will not cause the front air bags not to deploy during a crash event.
Sounds like the seat either does not contain the same pyro devices (belt pretensioner or side air bag) as the previous seat), bad wiring or not connected, different wiring, or the pyro device has already been used i.e. the new seat was in a car where the belt pretensioner deployed. Once a pyro device deploys it usually opens up electrically
Look at the pretensioner pyro (There should be an open ended 6" tube) if there is a cable end sticking out of the open end (open towards front of vehicle) the pretensioner has been used and needs to be replaced.
Get someone to scan the codes and it will tell you exactly what's up...
Dearborn Blue Oval