The celebrated theorem of Kechris, Pestov and Todorčević connecting structural Ramsey theory with topological dynamics has as a consequence that the Fraïssé limit of a Ramsey class of non-trivial finite relational structures has a reduct which is a total order; this implies an earlier result of Nešetřil, according to which the structures in such a class are rigid (have trivial automorphism groups). In this paper, we give an alternative proof of this fact. If $\mathcal{C}$ is a Fraïssé class of rigid structures over a finite relational language, then either the Fraïssé limit of $\mathcal{C}$ has a reduct which is a total order, or there is an explicit failure of the Ramsey property involving a pair $(A,B)$ of structures in $\mathcal{C}$ with $|A|=2$.