That's essentially it. The BFO simply injects the carrier that the transmitting station removed. By careful adjusting of this "missing carrier" frequency, the receiving station can change the Donald Duck sound into actual language.
Receivers that don't have BFOs, like your typical clock radio, or the older console radios from the 40s and 50s that had a shortwave band or two but didn't have a BFO, couldn't have clarified SSB by themselves. In many cases, such a radio can do it if you have a very stable external oscillator tuned to the proper frequency and loosely coupled to the receiver.