We consider asymptotics of power series coefficients of rational functions of the form 1/Q where Q is a symmetric multilinear polynomial. We review a number of such cases from the literature, chiefly concerned either with positivity of coefficients or diagonal asymptotics. We then analyze coefficient asymptotics using ACSV (Analytic Combinatorics in Several Variables) methods. While ACSV sometimes requires considerable overhead and geometric computation, in the case of symmetric multilinear rational functions there are some reductions that streamline the analysis. Our results include diagonal asymptotics across entire classes of functions, for example the general 3-variable case and the Gillis-Reznick-Zeilberger (GRZ) case, where the denominator in terms of elementary symmetric functions is 1-e1 +ced in any number d of variables. The ACSV analysis also explains a discontinuous drop in exponential growth rate for the GRZ class at the parameter value c = (d - 1)d-1, previously observed for d = 4 only by separately computing diagonal recurrences for critical and noncritical values of c.