  We provide infinitely many rational homology 3-spheres with weight-one
fundamental groups which do not arise from Dehn surgery on knots in $S^3$. In
contrast with previously known examples, our proofs do not require any gauge
theory or Floer homology. Instead, we make use of the $SU(2)$ character variety
of the fundamental group, which for these manifolds is particularly simple:
they are all $SU(2)$-cyclic, meaning that every $SU(2)$ representation has
cyclic image.
