“A cigar is sometimes just a cigar”, Freud. The original appearance is in 1919's edition of Atombau und Spektrallinien, not Wikipedia's 1924's, long before the "new" quantum mechanics. Bohr did wield his Zauberstab to import classical results into quantum physics, despite the inconsistency between the two, and to match the right answers from the spectral measurements. The context is discussed in Borelli's The emergence of selection rules and their encounter with group theory, 1913-1927:
"Using classical electromagnetism, Rubinowicz computed the angular momentum of a
spherical wave and showed that, when renormalized by $h/2π$, it had an absolute value equal or less than one. (Rubinowicz, 1918a, pp. 443-444). This led to a selection principle for the azimuthal
quantum number: $\Delta n = 0, ±1$ (Rubinowicz, 1918a, pp. 444-445)... It was left to Sommerfeld, in the first edition of Atombau und Spektrallinien (1919), to connect Bohr and Rubinowicz's results to his own interpretation of spectral terms as linked to
azimuthal quantum numbers. Sommerfeld showed a very clear preference for the theory of his
former assistant Rubinowicz, which he discussed at length (Sommerfeld, 1919, pp. 390-411).
Rubinowicz's "selection principle" was for Sommerfeld a means to bridge the gap between classical and quantum physics... He admitted that Bohr's condition $\Delta n = ±1$ fitted much better
the Rydberg-Ritz formula than Rubinowicz's, but made clear the epistemological gap he perceived
between a theoretically significant selection principle like Rubinowicz's, on the one side, and Bohr's
empirically successful condition, on the other. The latter, he described as a "magic wand"
("Zauberstab") to make quantum theory useful in practice (Sommerfeld, 1919, pp. 406-411, quote
from p. 402)".