So some time after posting my problem with the 2011 Macbook Air, above, I was giving it a long last look before tossing it.
(See my 'answer' to my own question, below, for the picture that relates to this explanation.) Next to the battery connector is a package about 3/4 inch square. On one side it attaches to the logic board by two stout solder points. Looking very closely I discovered cracks across both of these connectors. Wiggling this package would intermittently cause the battery to actually begin charging. So — solved. A touch with the soldering iron and a tiny bit of additional solder and the Macbook is back in business.
I would add, this computer has been severely abused. Dropped, hurled, knocked around. The case is so badly bent you can practically hold it edge-on and see daylight out the other side. My wife thinks I should send it to Apple as a testimony to its toughness. No idea how much longer it may last, but for now it is repaired and working as it should. At least as regards charging the battery.