FPRAS on #P complete problems and self reducibility
This theorem from that paper of Sinclair and Jerrum is slightly stronger than the sentence quoted in the question, and gives the fpras mentioned by Dyer:


  THEOREM 4.6. Let $R\subseteq \Sigma^*\times\Sigma^*$ be self-reducible. If there exists a polynomially time-bounded randomised approximate counter for $R$ within
  ratio $1 + O(n^\alpha)$ for some $\alpha\in\mathbb R$, then there exists a fully polynomial randomised approximation scheme for $\#R$.


Note the quantification over $R$. Your first question asks if "counting self-reducible structures might have an FPRAS", but this theorem is looking at particular $R$'s and showing that if $R$ has a weak kind of approximation algorithm, then $R$ also has a stronger kind of approximation algorithm.

Jerrum and Sinclair explain it like this:


  The chief significance of Theorem 4.6 is that it establishes a notion of
  approximate counting which is robust with respect to polynomial time
  computation, at least for the large class of self-reducible relations [...] We suggest that this notion will be useful in the future classification
  of hard counting problems [...]