Any uniqueness to your climate or location? I see a lot of green in one of your photo backgrounds, could this be built up pollen and dust bound by minerals from the rain? Maybe combined with the residue from other cleaning attempts.
I would try a bristled scrubbing brush with dish soap first. If that didn't work, I'd try the brush with a weak vinegar to see if it's a pH issue.
Another thought: this might be plastic coating on the mesh degrading after extended UV exposure. In which case you probably have to replace it, but might as well try some home chemistry first?