Felt like connecting my GameCube again, and I remembered that the controller input was iffy last time I tried. Bought a new third-party controller and plugged it all in. Didn't seem to work. Looked for my old controller, but it's missing. Went and bought a used first-party controller, plugged it in.
Turns out it wasn't the controller which was iffy, it was the controller jacks in the base unit.
I'll buy a replacement unit next time I'm out, but I think the ¥3500 I've spent on controllers is what it will cost to get an entirely new base unit. I'd have done that to begin with, had I known... :'(