Can the theory of the second best mathematically justify labor unions in some scenarios?
Quoting your quote, emphasis altered by me:

The “theory of the second best” clearly argues that once markets depart at all from perfect competition, efficiency may well be increased by further departures. For example, in the case of monopsony power in low-wage labor markets, legislated minimum wage increases can potentially move wages closer to efficient levels and increase employment.

It seems to me the quoted Economist summary misrepresents this quite a bit! No one is claiming that unions or state intervention will yield the second best (which is a somewhat vague concept - what are the exact information constraints here?), merely that they may be welfare increasing.
To expand on this a bit: if we don't know where &quot;the competitive equilibrium&quot; wage would be, but people agree that it is in the [8,40] dollar range, then instituting an 8 dollar minimum wage policy would not have a detrimental effect and would be welfare improving in cases when the current wage is below \$8. But there is no claim of optimum, not even in expected value.
Mathing it up:
$$
\begin{align*}
S(p) &amp; = p \\
D(p) &amp; = a - p \ \text{ where } \ a \sim \text{U}[16,80]
\end{align*}
$$
so
$$
p^* = \frac{a}{2} \sim \text{U}[8,40].
$$

The paragraphs above are the gist of the answer, but I can also produce a hiperbole on par with the Economist. Suppose that I told you that due to variance in quality some lollipops manufactured by a certain company are detrimental to one's health. The Economist seems to argue, that someone proposing the idea of quality control is promising not that they will catch some of these poisonous lollipops but all of them. And since this is clearly not possible, as quality control itself has varying quality, it is pointless to even try to improve the situation. But quality control may be justified even if some of the subpar lollipops are discovered. In fact quality control may be justified even if it supervised by a nonexpert and hence only catches half as many subpar lollipops as possible, as stopping some bad effects are preferential to stopping none.
It occurs to me that given the difference between the marital ideal and the status of the average marriage it is rather fortunate The Economist does not give marital advice.