From your first paragraph I'd be for checking your battery leads but I see you're on to that. The positive lead can corrode under the insulation to the point where it may show volts OK but doesn't like to carry current (amps). Earth can be the same but the fault should be more visible. Often mentioned is the possibility of damage to the positive lead where it goes under the front of the engine but you'll see evidence, if any, of that when you do the replacement.
Do the early cars have the small earth lead from the back of the head to the bulkhead? That's useful backup. Clean back any earth connections at the body end.