  We propose a new functional analytic approach to Stein's method of
exchangeable pairs that does not require the pair at hand to satisfy any
approximate linear regression property. We make use of this theory in order to
derive abstract bounds on the normal and Gamma approximation of certain
functionals in the Wasserstein distance. Moreover, we illustrate the relevance
of this approach by means of three instances of situations to which it can be
applied: Functionals of independent random variables, finite population
statistics and functionals on finite groups. In the independent case, and in
particular for symmetric $U$-statistics, we demonstrate in which respect this
approach yields fundamentally better bounds than those in the existing
literature. Finally, we apply our results to provide Wasserstein bounds in a
CLT for subgraph counts in geometric random graphs based on $n$ i.i.d. points
in Euclidean space as well as to the normal approximation of Pearson's
statistic.
