Hello,
I see two explanations:
1) the vertical velocity with hydrostatic assumption is only a cosmetic by-product of horizontal velocities, obtained with the continuity equation and used only for drawing pictures (there is one however for advection in the transformed mesh, but also obtained with the continuity), it is not so good if the vertical velocity is not so small...
2) the hydrostatic assumption is wrong on weirs, and the theory leads to a unique capacity of the weir, independently on the shape of the weir, whereas all hydraulic handbooks will give you different values function of the weir shape. This is because the vertical velocity is neglected.
The latter is a strong argument...
With best regards,
Jean-Michel Hervouet