You should clean the battery contacts anyway. I didn't have the problem, but I heard, that some other guy had a completely non-working 71, which worked after a battery contact cleaning. Good contacts seem to be rather important, especially, when the 71 hasn't been in use for some time.
There might be a contact problem, even, if you don't see any dirt or corrosion on the contacts. The 71s battery contacts have been changed some times in the life cycle of the 71, so I'm sure HP was aware of this. I own a first series 71, which doesn't have the problem, maybe just because I'm using it since I bought it from HP.
You should try using the 71 some hours with the mains transformer. If this works without problems, you have isolated the cause.
I don't believe, that some electrolyte capacitors are bad, because the batteries provide a really large capacitance to the machine. Normally capacitors could be bad, if the machine works with the batteries, bot not with the mains supply. That case could be dangerous to other calculators, but not to the 71. The 71 can be driven without batteries inside. It has a rectifier and some capacitors inside.
When you have further problems with the mains supply attached, you possibly have a broken flex PCB (printed circuit board) between the main (lower) PCB and the keyboard (upper) PCB. That would be bad. I could fix it for you just for shipping costs (take this offer as a Christmas gift) :-) . You could also find an electronics engineer, who opens the 71 and repairs (replaces) the flex PCB with some wires. I can supply further information with email, when neccessary.