I have found the "Spray Nine" AP disinfectant cleaner is amazing for what it can remove. Although not specifically designed to do a whole deck, I would probably put it in a pump up sprayer to apply and then brush/agitate. The fact that it is rated as a disinfectant is just a bonus, it just cleans fantastically. I use it for many applications. Anyway, just a thought for you; not too expensive to try a bottle.
https://www.amazon.com/gp/product/B0...?ie=UTF8&psc=1