Why does the Co³⁺/Co²⁺ couple have such a high reduction potential?
The electronic configuration has nothing to do with it. The reduction potentials of $\ce{Ni^3+}/\ce{Ni^2+}$, $\ce{Cu^3+}/\ce{Cu^2+}$ and $\ce{Zn^3+}/\ce{Zn^2+}$, if they have been/could be measured, would be even greater.

The reduction potential for $\ce{M^3+}/\ce{M^2+}$ is most dependent upon the third ionisation energy. If $I_3$ is large then it will be difficult to oxidise $\ce{M^2+}$ to $\ce{M^3+}$ (hence a high reduction potential). Likewise if $I_3$ is small then the reduction potential will be smaller.

$$\begin{array}{c|cc} \hline
\text{Metal} &amp; I_3\text{ / eV} &amp; E^\circ(\ce{M^3+}/\ce{M^2+})\text{ / V} \\ \hline
\ce{Sc} &amp; 24.76 &amp; (-2.6) \\
\ce{Ti} &amp; 27.48 &amp; (-1.1) \\
\ce{V}  &amp; 29.31 &amp; -0.26 \\
\ce{Cr} &amp; 30.96 &amp; -0.41 \\
\ce{Mn} &amp; 33.67 &amp; +1.60 \\
\ce{Fe} &amp; 30.65 &amp; +0.77 \\
\ce{Co} &amp; 33.50 &amp; +1.93 \\
\ce{Ni} &amp; 35.16 &amp; (+4.2) \\
\ce{Cu} &amp; 36.94 &amp; (+4.6) \\
\ce{Zn} &amp; 39.72 &amp; (+7.0) \\ \hline
\end{array}$$



Data from Weller et al. Inorganic Chemistry 6ed; Johnson Some Thermodynamic Aspects of Inorganic Chemistry. Numbers in parentheses indicate predicted values. Graph by me in Microsoft Office 365.

So your question is essentially: why is $I_3$ of $\ce{Co}$ so "large"?

Going across the 3d block, effective nuclear charge increases which leads to the general trend of IE3 and $E^\circ(\ce{M^3+}/\ce{M^2+})$ increasing.

Any stabilisation derived from "special" electronic configurations e.g. the half-filled subshell in $\ce{Fe^3+}$, or the ligand-field stabilisation energy of $\ce{Cr^2+}$, merely lead to relatively small deviations from the general trend.

Moral of the story is: look at the big picture first. Those $\mathrm{d^5}, \mathrm{d^{10}}, \cdots$ configurations can be relevant, but they aren’t responsible for the overall trend, and certainly shouldn’t be the first thing you consider when trying to rationalise this set of data.