I think I figured it out. Evidently the layer was turned "off" even though the path itself showed up as selected. If I click the checkbox to toggle the layer on and off it makes no visual difference to my path or the active layer below it... i.e, there's nothing to warn me that the path is in an "invisible" layer. AFAIK this is how the layer created itself when I clicked on the pen tool and started "drawing" with it. Maybe I accidentally hit some key combo that turned "off" the layer?
When I click on the checkbox to "show" the layer, however, the "Selection" button does what I expected all along. (Other than the fact that it deletes the path, which apparently is how it was designed!) That is, the selected path turns into a pixel selection.