Solved! Go to Solution.
The bridge "surface" is curvature continuous in that if you create a section perpendicular anywhere along the joining edge it will show a smooth transition. However the side "edge" of the bridge does not by default stop and run smooth to the side edges of the input faces. in this case it is more than a degree sharp.
Usually to get the bridge side edges smooth you would change the Flow Direction to Iso Parametric, forcing the isocurves of of the bridge to match the isocurves of the input faces and thus the side edges will be smooth (assuming natural trimmed surface). In this case you cannot do that as the surface goes to a point.
The best solution in this case is to forget the trim and bridge features. Edit the Blend to Shape - Curvature and change to trim and sew.
Not sure why the tangent select is not working with the through curve command, but if you manually select the function takes the edges as tangent, they appear clean enough edges.
Interestingly tangent select works in other functions such as fill surface.