I think you misunderstand what syncing does and does not do, I sync my calendar as a back up but it would not stop overlapping booking on multiple sites. Seems to happen daily with me. None of them work instantaneously and in this case it was not AirBnB but the other system that did not update.
You can certainly use IB on AirBnB but if you have the equivalent elsehwere you will get double bookings. So only one of your systems can use IB.
If you get an IB on AirBnB take those dates out of the other systems, if you get a request elsewhere before accepting block those dates on AirBnB. I know it happens with Hotels, but does not matter so much to them as they have multiple rooms, most AirBnB hosts do not have that luxury.
David