Let's hope the batteries make a difference, but I agree that it would be odd that they work fine in one camera and not in another. However, stranger things have happened!
Sort of; firstly, it's a bit of an educated guess on my end, so no guarantees that this is really the problem. And even if it is, the bad news is that it'll be far more difficult to resolve than on the far more simple cameras that are often associated with capacitor problems. These EOS cameras are highly integrated and the electronics are very complex. This means that identification of potentially bad capacitors is already a major challenge, as well as the disassembly involved in it.
All considered, if the batteries don't help, things will likely get very complicated, very quickly.