I use a percarbonate base cleaner & brighten with a oxalic acid.
Works great. Also Wohmans deck stripper works well and it can be brighten with oxalic acid also.
Let your chemicals do the work. Inspect inside the attic before and after the job. Always walk sideways, never stepping on two different rows of shakes at one time. Use corkers or track shoes with spikes to walk on roof & wet roof down before walking on it, this helps prevent splitting of shakes when walked on. Clean gutters out at end of job-they are bad to get stop up with wood fibers, moss, gunk, pieces of shakes etc. Be prepared to replace shakes, felt, and caulking.
Don't just clean the roof, restore it.
PS Shakes + Moss + Chemicals + water = Ice Very Slippery.
If the roof is not sealed - in my opinion, the customer is wasting their money.