Massively parallel systolic arrays and resource-efficient depthwise separable convolutions are two promising techniques to accelerate DNN inference on the edge. Interestingly, their combination is inefficient: Computational patterns of depthwise separable convolutions do not exhibit a rhythmic systolic flow and lack sufficient data reuse to saturate systolic arrays. We formally analyse this inefficiency and propose an efficient operator, an optimal hardware dataflow, and a superior training methodology towards alleviating this. The efficient operator, called FuSeConv, is a drop-in replacement for depthwise separable convolutions. FuSeConv factorizes convolution fully along their spatial and depth dimensions. The resultant computation efficiently maps to systolic arrays. The optimal dataflow, called Spatial-Tiled Output Stationary (ST-OS), maximizes the efficiency of FuSeConv on systolic arrays. It maps independent convolutions to rows of the array to maximize resource utilization with negligible VLSI overheads. Neural Operator Scaffolding (NOS) scaffolds the training of FuSeConv by distilling knowledge from the expensive depthwise separable convolutions. This bridges the accuracy gap between FuSeConv networks and baselines. Additionally, NOS can be combined with Neural Architecture Search (NAS) to trade-off latency and accuracy. The HW/SW co-design of FuSeConv with ST-OS achieves a significant speedup of 4.1-9.25X with state-of-the-art efficient networks for ImageNet. The parameter efficiency of FuSeConv and its significant out-performance over depthwise separable convolutions on systolic arrays illustrates their promise as a strong solution on the edge. Training FuSeConv networks with NOS achieves accuracy comparable to the baselines. Further, by combining NOS with NAS, we design networks that define state-of-the-art models improving on both accuracy and latency on systolic arrays.