All-carbon field-effect transistors, which combine carbon nanotubes and graphene hold great promise for many applications such as digital logic devices and single-photon emitters. However, the understanding of the physical properties of carbon nanotube (CNT)/graphene hybrid systems in such devices remained limited. In this combined experimental and theoretical study, we use a quantum transport model for field-effect transistors based on graphene electrodes and CNT channels to explain the experimentally observed low on currents. We find that large graphene/CNT spacing and short contact lengths limit the device performance. We have also elucidated in this work the experimentally observed ambipolar transport behavior caused by the flat conduction- and valence-bands and describe non-ideal gate-control of the contacts and channel region by the quantum capacitance of graphene and the carbon nanotube. We hope that our insights will accelerate the design of efficient all-carbon field-effect transistors.