We revisit the primitive relay channel, introduced by Cover in 1987. Recent work derived upper bounds on the capacity of this channel that are tighter than the classical cutset bound using the concentration of measure. In this paper, we recover, generalize, and improve upon some of these upper bounds with simpler proofs using reverse hypercontractivity. To our knowledge, this is the first application of reverse hypercontractivity in proving first-order converses in network information theory.