I don't know if it was an endemic problem or if it was just something that
happened to the one we had, but our Odyssey had some really nasty
corrosion on the sockets and PCB traces for the external connectors
(video, controllers, etc.). It almost looked like they had used acid-core
solder by accident or something. We cleaned it all up, resoldered the
connections with rosin-core solder, and installed a jumper to replace
one of the damaged traces, and that fixed it right up. I don't remember
if it was totally dead before that, or if it just had some weird symptoms,
but it's something to check, if you haven't already.