I have a lightweight bike that shimmies too. I only noticed it after swapping the wheels. But my guess is that it shimmied with the previous wheels too — I just hadn’t tried riding no hands at speed. I do think the shimmy is more pronounced the further back I am on the saddle. My guesses: frameset resonance, unbalanced wheel weight, tires and/or wheels not mounted perfectly straight; replacement fork/geo. Also maybe a bent frame, but IIRC mine passed the string test too. Shimmy is quite the mystery.