An outside breeze does not make a lot of difference with natural convection in the attic. I can imagine that even if there were a good outside breeze, if you were to put your face near the upwind gable, the heat might still drive hot air out that gable, still overcoming the slight pressure of the wind.
Most comments here are telling you that passive natural convection from cooler air low in the attic to the hot air along the ridge or gables will keep the roof shingles from baking. You goal is not so much to have a "cool" attic but a cool living space with minimum heat gain from the attic.
Perhaps more concerning is that the lower attic is cooler because there is more heat transfer from there to the living space. That could be from less insulation between it and the living space adjacent to it (below and beside), or there is cool air leakage into that attic. While that makes the attic cooler, not for a good reason.
Maybe the main problem is inadequate or blocked soffit vents. A passive solution is the best approach...it works even without power.
Consider this: There is not a lot of difference between the heat gained by the second floor living space through the ceiling when the attic is 120F and the bedrooms are 68F versus 110F and 68F. (52F difference versus 42F). Try to see if insulation was blown out over the soffit vents, or if you need more of them.
