It appears that if you draw a tangent to your glider's
polar beginning, not at zero, but at any given headwind
speed, the line will touch the polar at a point that
is best L/D plus half that headwind.

Close to it but, yes - as you can see in the lines drawn on the L23
polar of the article
http://home.comcast.net/~verhulst/GB.../headwind.html

I was under the impression it was added to give you a margin for gusts
and turbulence, which are usually less than the average wind speed. The
"half" was likely chosen empirically, as something that was adequate
almost all the time.

I suspect that you may be confusing the "best speed to cover the most
ground in a headwind" with "the best speed to make a safe approach to
landing".

Tony V.