Options would be to either move the passenger to another seat & placard that seat as unusable, or go back to the gate and have maintenance fix what's broken. In the former case, the seat would most likely be fixed by a mechanic at the next station.
The seatbelts used in the F/A demos are seatbelt extensions, which still need the installed seatbelt itself to be intact. They are fully functional as such, though, unlike the demo life vests & O2 masks, which indeed are "demo only" and of no use at all in a real emergency.
For a mechanic, the seatbelt is quick & easy to replace, although without needle-nose pliers (and a spare male or female belt), rather difficult for the crew -- who wouldn't be allowed to anyway.
The FAA requires that every passenger has a seatbelt and has been told how & when to use it, and that the crew has verified that they have it on before takeoff.