Quasimodular forms that detect primes are Eisenstein

MacMahon's partition functions and their extensions provide equations that identify prime numbers as solutions. These results depend on the theory of (mixed weight) quasimodular forms on $SL_2(\mathbb{Z})$. Two of the authors, along with Craig, conjectured an explicit description of the set of prime-detecting quasimodular forms in terms of Eisenstein series and their derivatives. Kane et al.\ recently verified this conjecture using analytic methods. We offer an alternative proof using the theory of $\ell$-adic Galois representations associated to modular forms.