I absolutely think it's instructive to view the LCM as a specific case of meets in a semilattice, and in turn to view these as specialization of a general theor

I absolutely think it's instructive to view the LCM as a specific case of meets in a semilattice, and in turn to view these as specialization of a general theory of categorical products. (Why wouldn't it be instructive? If you like, the "unusual algebraic structure" on the natural numbers which might be worth considering is that of a category with products). And I also think that recognition of such analogies and general patterns ubiquitous across mathematics is a large part of the spirit of category theory.