The airline can move passengers around, even if they have chosen seats (you'll always see the disclaimer when you choose a seat online) - whereas the online booking system won't necessarily know if it can do this.
Also, the airline may have more up to date knowledge of cancellations at that stage.
So the problem isn't the airline making it look like there are few remaining seats, but that the system won't let you move other passengers in order to fit a group in at short notice.
When I make late bookings with the family, I just plan for the fact that we may all have to sit in different areas on the plane. Sometimes it can be sorted, but certainly not every time. That's the hazard of late booking.