Here's what Canon tells me: my 1DsMk3 power supply had a short. A printed circuit board had to be replaced. That is outside the functions Err80 usually reports on, but "cards and control" obviously suffered in the power issue. <grin>
Graham, your idea about trying EOS Utility to bypass cards as a diagnostic was very interesting. I do use tethering a lot. Would not have shown us the problem here.
The shutter count survived the 48-hour button-battery-less interlude.
Canon had it out the door the second day (at Irvine, California, on a Friday later morning and out with FedEx on Monday afternoon). They also cleaned the sensor and the viewfinder, gave me a shutter count. I wish they'd write the count down so I can show it to any future purchaser, but they told me over the phone. (Second time they didn't log it in writing but I am pleased by this service. Reasonable cost - especially with CPS discounts.
Thanks, all, for ideas.