The subsurface structure of an "average" supergranule is derived using existing data products from the Helioseismic and Magnetic Imager (HMI) time-distance pipeline and compared to the best helioseismic flow model detailed by Duvall and Hanasoge ( Solar Phys. 287, 71, 2013). We find that significant differences exist between them. Unlike the shallow structure predicted by the model, the average HMI supergranule is very extended in depth, exhibiting horizontal outflow down to 7 - 10 Mm, followed by a weak inflow reaching a depth of ≈ 20 Mm below the photosphere. The maximal velocities in the horizontal direction for the average supergranule are much lower than in the model, and its near-surface flow field RMS value is about an order of magnitude lower than the often-quoted values of ≈ 250 - 350 m s-1 for supergranulation. Much of the overall HMI supergranule structure and its weak flow amplitudes can be explained by examining the HMI pipeline averaging kernels for the near-surface inversions, which are found to be very broad in depth, and nearly identical to one another in terms of sensitivity along the z-direction. We also show that forward-modeled travel times in the Born approximation using the model (derived from a ray-theory approach) are inconsistent with measured travel times for an average supergranule at any distance. Our findings suggest systematic inaccuracies in the typical techniques used to study supergranulation, confirming some of the results of Duvall and Hanasoge ( Solar Phys. 287, 71, 2013).