I'd recommend using a very strong glass cleaner because I don't really think it's the wipers.
I think the windshield is coked with road grime.
regular car washing doesn't remove all of it.
You also can't see it either, but it's there.
I've used Nu-Finish car polish on my glass and it removed the grime, and does not scratch/swirl the glass.
You could also use a non gel type toothpaste, like Colgate.