Holy crap! After all that trouble, you're right, it was just physical damage to the pins. There's nothing visible, but a little wiggling with a safety pin to push the clippy bits closer together, and now it works. Hooray!

I guess this means the little USB gadget is OK, provided I don't use it too much and I bring a safety pin with me when I travel. If I keep wiggling the pins like that they'll wear out eventually, so it's just a question of how much the portability is worth to me in replacement battery costs.

Thanks for the help, and I definitely learned some stuff :-)