I agree with Rob, I would like to help all customers but sometimes there is a situation where the downside might cost more than the profit is worth on the job.
No idea which acid might, might work, you might have to try several until you find the right one that works.
I think that CLR and Limeaway work on hard water stains and calcium deposits but not sure what that would do to a roof, the flashing, the nails, etc.......
Not sure what shape the actual shingles are in, it might be close to the time to re-roof the house.
Just too many variables for me to get into this, the downside is too big if I was there.
Something else to think about is what if the acid you use had adverse affects on the gutters, plants, wood, siding, shingles, etc.... that nobody thought would happen and in a few months you get a letter from their attorney wanting you to replace the damaged property (part of the roof and some other minor things), would the money for the experimenting to get the previous roof cleaner residue off along with the roof cleaning pay for a new roof and replace the minor things? ? ? ?
You never know, this could be a setup for a new roof.
This is one of those bad "Gut Feeling" kind of jobs that you want to just walk away from before anything gets blamed on you.
If this was a botched up concrete job where it was just wand marks or if someone tore up some wood like used a surface cleaner on it or the wrong pressure, you can go and spend time and labor there and make it look better with your experience. It is not a critical part of the house like a roof where if a leak started, it could be easily blamed on you, there could be interior damage (walls and ceiling need to be re-painted or replaced), water leaked onto their piano, computer or other expensive household items that now need to be repaired or replaced.
I am not even in this city but have that bad gut feeling about this job.
Just my opinion.