Bill, my contractor did not look at the roof. I told him it was leaking and he said "you need a new roof." He said he had his carpenter climb up to see how many layers were up there (one) but I didn't see him do it. I'm just back into town and will climb up tomorrow and look.
I can see the sky from between the central chimney and the roof. My beginner's guess is that the shingles are fine but whoever put them on did a lousy job - no flashing around the chimney and probably none around the vent. The water spot was not near the chimney and there are water marks beginning to show through the ceiling in my bedroom which was renovated right before I bought the house.
I'm concerned about just filling in the gaps. If they did a poor job to begin with, doesn't that make the expected life of the shingles moot?