The number is not arbitrary, and it did not arrive with this system. It has weight in several traditions the compilers knew.
The Letter of Aristeas, a Hellenistic Jewish text, has seventy-two elders, six from each of the twelve tribes, produce the Greek translation of the Torah, which is why the Septuagint carries a name meaning seventy despite the story specifying seventy-two.
The table of nations in Genesis 10 was counted as seventy in the Hebrew tradition and seventy-two in the Greek, which gave rise to the long-standing idea of seventy or seventy-two nations and seventy or seventy-two guardian angels over them.
The sending out of the disciples in Luke 10 reads seventy in some manuscripts and seventy-two in others; the variant is ancient and is noted in modern critical editions.
The elders of Israel in Numbers 11 are seventy, with two more who prophesy in the camp, and the count has been read as seventy-two on that basis.
Seventy-two is also five degrees of a circle, six times twelve, and the product of eight and nine, the last of which is exactly the division used for the choirs.
So the number arrived already loaded, and the seventy-two-fold name landed in a tradition disposed to find it significant. That is worth knowing in both directions. It explains why the derivation was received with such enthusiasm, and it warns against treating the number's recurrence elsewhere as confirmation. A number that appears in this many places will appear in one more, and the appearance proves nothing on its own.