Just to clarify -- I power wash my plain flat surfaces and metal toe rails once a year. It does quick work of the green stuff, dirt in tight areas, etc. But I wound up blasting off the light gray paint from my non-skid areas, so stopped doing that. Which is why I asked about cleaners, wanting something that won't hurt the gray paint on the non-skid deck.
Latest result -- sprayed Totally Awesome directly on the black spots (no water), and let it sit about 10 minutes. It started turning to orange, which concerned me that a chemical reaction may be happening with the paint. Immediately flushed with lots of water, with no apparent damage to the paint. But the orange color change puts me off from leaving Totally Awesome on the surface for very long. Unfortunately, the black spots seemed to shrug it off. No change.
Next tried Comet powder on other spots, added a little water to form a paste. Let sit for a minute. Scrubbed off with brush. Some spots came off, some didn't. I still think its mainly the mechanical brushing that is having the most effect.
Still didn't have Tilex available. And the weather was turning nasty. So quit for the weekend. Still on the look out for that miracle cure. As I have learned over many years, one sailor's cure may not work for every case. But I'm not giving up, so keep sending ideas. I just remind myself that even scrubbing the deck on my boat has curative power to my mental state.
Thanks everyone. When I posted this mundane question, I didn't expect any answers at all. Nice to have an active group to learn/share with.