Weed definitely don't "prove" a lack of anything. They might suggest it, but definitely not prove it. The soil test was the proof.

Unless you have a way to grind them up very finely, I wouldn't just sprinkle them on a lawn. The edges are pretty sharp, so they don't exactly make a great walking surface. If you can grind them up small, sprinkle away. If not, compost them first and then spread the compost around. They will break down faster in a compost pile anyway.