I recently ran into a very similar problem with a white mulberry. I had thought it was a bacterial blight, since the spots on the leaves matched what I saw online. I treated it with a product advertised to treat bacterial blight, but the problem persisted.
At a local club meeting, I showed the tree to a Longwood Gardens curator, who pointed out the tree actually had scale insects. They were too tiny for me to discern the shape, and I had thought they were just lenticels. I treated the tree with both topical and systemic insecticides, and now it's recovering, pushing new growth.
Anyway, my point is that tree diseases are pretty tough to diagnose, and on top of that, one ailment will often invite other ailments of different types. There's a reason it's common to see three-in-one insecticide, miticide, and fungicide.