They show up in a few other areas, like probabilistic analysis of algorithms or sketching (e.g. Chernoff bounds and definitely the Cauchy-Schwartz ineq.). But you're right, nothing heavyweight like Muirheads has many applications.

But a surprising number of them can be solved with the insight only that x^2 >= 0. (Both Cauchy-Schwartz and AM/GM reduce to that if you squint hard enough). And that's accessible to any maths student.