Ultrafast photodetectors based on two-dimensional materials suffer from low responsivities and high dark currents. Interlayer exciton dissociation in type-II vertical heterojunctions of transition metal dichalcogenides is a viable mechanism for achieving higher responsivities with picosecond response times. Here, we propose a novel device concept based on these structures, with potential for self-powered photodetector applications characterized by an unprecedented trade-off between speed and responsivity with zero dark current. In order to assess the realistic performance to be expected in the proposed device, we have purposely devised a simulation approach able to provide a detailed investigation of the physics at play, while showing excellent predictive capabilities when compared with experiments on interlayer exciton transport available in the literature. The proposed high-performance photodetectors with tunable responsivities are at reach with available fabrication techniques and could help in paving the way towards monolithically integrated artificial neural networks for ultrafast machine vision in speed sensitive applications.