Transmsission engineers have dreamed of the perfect belt drive CVT for years ... you are trying to control a steel on steel friction interface between belt and two sets of variators. Both parts have to be heat treated to very high hardness and the variator surfaces need to be ground and polished to a tightly controlled surface finsih specificaiton. You put a very high pressure pump on the transmission and use hydraulc pressure to set clamping force and always make sure that you don't undershoot on force because the slightest slip of the belt destroys the variator surface. My opinion, the perfect belt drive CVT doesn't exist. The only CVT I care for is the hybrid powersplit designs used by Ford and Toyota as they use a gear set and electric generator to set ratio.
My first rule is real automatic transmissions have a torque converter, at least one one way-clutch, and transfer torque via planetary gear sets.