It does not seem from MacMahon's first mention of plane partitions that the $\Omega_\leq$ operator was relevant. At the end of Article 42 of his paper "Memoirs on the theory of the partitions of numbers---Part I", MacMahon says "This partition may be termed 'graphically regularised' by reason of its origination in a subjacent succession of lines in the bipartite graph. This species of regularisation is the natural extension to three dimensions of Sylvester's graphical method in two dimensions." He then goes on to develop some simple properties of plane partitions (without using that terminology) and to conjecture his famous generating function $\prod_{n\geq 1}(1-x^n)^{-n}$. He also suggests less confidentally that three-dimension partitions have the generating function $\prod_{n\geq 1}(1-x^n)^{-{n+1\choose 2}}$ (now known to be false). The $\Omega_\leq$ operator is used implicitly to prove some simple results, but it does not seem to be relevant to MacMahon's original motivation. I believe that MacMahon did not explicitly use his $\Omega_\leq$ operator until "Memoirs on the theory of the partitions of numbers---Part II", about three years after Part I. In Part II he does consider plane partitions as an example.

It does not seem from MacMahon's first mention of plane partitions that the $\Omega_\leq$ operator was relevant. At the end of Article 42 of his paper "Memoirs on the theory of the partitions of numbers---Part I", MacMahon says "This partition may be termed 'graphically regularised' by reason of its origination in a subjacent succession of lines in the bipartite graph. This species of regularisation is the natural extension to three dimensions of Sylvester's graphical method in two dimensions." He then goes on to develop some simple properties of plane partitions (without using that terminology) and to conjecture his famous generating function $\prod_{n\geq 1}(1-x^n)^{-n}$. He also suggests less confidently that three-dimension partitions have the generating function $\prod_{n\geq 1}(1-x^n)^{-{n+1\choose 2}}$ (now known to be false). The $\Omega_\leq$ operator is used implicitly to prove some simple results, but it does not seem to be relevant to MacMahon's original motivation. I believe that MacMahon did not explicitly use his $\Omega_\leq$ operator until "Memoirs on the theory of the partitions of numbers---Part II", about three years after Part I. In Part II he does consider plane partitions as an example.