The FMGC flys the airplane. FMGC = master computer in simple terms
By severe stretch of imagination, it just might be true. FMGS is integrated FMS/FD/AP/Autothrust. It calulates navigation and guidance parameters for FD, sends commands to FCCs and FADECS via AP and ATHR respectively. Kick out the AP and ATHR and FMGS has no authority over aeroplane anymore. ATA22.
The pilot communicates with the FMGC through the MCDU - keyboard that programs, the FCU - glareshield mounted knobs and buttons used to request specific speeds/altitudes/headings/vertical speeds- and the SS.
You are wrong on sidesticks - they are not connected to FMGC but rather to flight control computers - ELACs/SECs on minibus, PRIMs/SECs on widebodies. ATA27.