Wall damage from insufficient attic ventilation
Your problem , as you noted, pertains to moisture. Specifically, moisture laden air coming from the living space below via small gaps and cracks (ie. conduit and wire runs, vents and register openings cut to large, electric boxes and outlets openings cut wider than needed, window door openings not sealed around the rough opening, there's a few more I can't think of right now).
There's also the issue of attic insulation. The original powdered stuff is a very poor insulator. It won't prevent the heat from the attic or the cool from the living space from transmitting (radiating?)    to and from the attic. If I recollect correctly, if warm air hits the cool bricks condensation will form on the bricks in the attic. Maybe not enough to be visible, but enough to be drawn into a porous red brick. I think you know the rest: the moisture contains salts from the brick and mortar, when it evaporates mineral deposits (efflorescent's) is left behind.
The repair is to -seal any passage way's from air infiltration below. Use expanding foam and/or any caulk to fill openings.
-insulate the attic floor. A better but bigger job is to insulate the attic rafter bays. I see HVAC ducts in your attic. Even though insulated the summer heat makes them less efficient at moving/keeping air cool. A radiant foil barrier, I found, works extremely well at reflecting heat from the exterior back out through the roof.
-seal/paint the brick. A clear masonry sealer and/or a masonry paint seals the porous bricks with a moisture impervious barrier. Clear seal on the living space side and dryloc on the attic side.
-an extra turbine closer to the problem area would help as a last additional solution.
Finally, just so you know the powder residue is not a serious problem. That is to say it's not a detrimental fault that over time will effect the integrity of your home. See it as an aesthetic issue.
Lastly, in my opinion the framing, raised pink insulation and vent location and other barriers are not causing this problem.