OP's solution posted as an edit to the question:
Here is what I ended up doing. Disclaimer I am definitely not an expert. I also had white spots that were likely mold. I did not notice/find any water leaks, the house is 70 years old. So I'm assuming some mold has just grown over time. I read bleach is not necessarily good for mold removal on wood. So I decided to try other chemicals. I've mixed 3% hydrogen peroxide (store bought) half and half with water in a spray bottle. Sprayed down, let sit for at least 10 min, scrubbed off (wearing p100 respirator, eye mask, and gloves), rinsed with water, and dried off. So far this looks to have effectively removed the mold and the patterns seen in the original image.
In terms of preventing, and getting some mold in difficult spots, I may try Concrobium to kill and prevent further growth. Alternatively, I also have basic Borax that I might mix into a solution and apply after my hydrogen peroxide to try to prevent further growth. Bora Care with Mold Care (which actually contains some of the same ingredient as Borax) sounds like a good product to apply too, but is quite expensive and might be overkill for my situation.