I don't understand your interpretation of what Dobrushin actually did. Let me restate it in terms closer to his work (as it was the language of functional analysis that he used). A family of probability measures $\pi_x$ on the space $Y$ indexed by points $x\in X$ provides a linear operator ($P$ in Dobrushin's notation, and $W$ in yours). Then he notes that if one endows spaces of measures on $X$ and $Y$ the the total variation norm, then the norm of the restriction of $P$ to zero total mass measures is precisely $\sup_{x_1,x_2\in X} \| \pi_{x_1}-\pi_{x_2} \|/2$. Your contraction coefficient is 1 minus this norm (formulas 1.5, 1.5', 1.5''', 1.15). Therefore, in order to realize it one should look at singular (rather than total variation close) pairs of measures on $X$.

EDIT: If there are two probability measures $\mu_1,\mu_2$ on $X$ which are **not** mutually singular (i.e., $\|\mu_1-\mu_2\|<2$), then there is a uniquely defined "common part" $\lambda$ such that $2\|\lambda\| + \|\mu_1-\mu_2\|=2$ and $\mu_i=\lambda +\mu'_i$, where $\mu'_1,\mu'_2$ are now mutually singular. Let $\overline{\mu'_i}=\mu'_i/(1-\|\lambda\|)$ be normalizations of $\mu'_i$. Then obviously
$$
\| P\mu_1 - P\mu_2 \| = \| P\mu'_1 - P\mu'_2 \| = (1-\|\lambda\|) \|P\overline{\mu'_1} - P\overline{\mu'_2} \| \;,
$$
or
$$
\|P\overline{\mu'_1} - P\overline{\mu'_2} \| = \frac1{1-\|\lambda\|} \| P\mu_1 - P\mu_2 \| > \| P\mu_1 - P\mu_2 \| \;,
$$
so that in order to maximize $\| P\mu_1 - P\mu_2 \|$, the measures $\mu_1,\mu_2$ **must** be singular.

PS Of course, without any additional assumptions on the family $\{\pi_x\}$ there is no reason to think that $\sup_{x_1,x_2\in X} \| \pi_{x_1}-\pi_{x_2} \|$ is attained. For instance, one can easily construct a family of pairwise absolutely continuous measures (so that no two of them are mutually singular) with $\sup_{x_1,x_2\in X} \| \pi_{x_1}-\pi_{x_2} \|=2$.

PPS I use here the literal definition of the total variation norm, so that $\|\mu\|$ is the sum of the masses of the positive and negative parts of the Hahn decomposition of $\mu$. Unfortunately, what many probabilists (apparently, in the belief that in probability there is no place to numbers greater than 1, and wanting to set probability apart from the rest of mathematics) call "total variation distance" is the result of dividing the true total variation by 2. Alas, it is this misleading definition that appears in the wiki article http://en.wikipedia.org/wiki/Total_variation_distance_of_probability_measures. For the sake of historical truth one has to admit though that Dobrushin himself **did** divide by two in the quoted paper.