I wanted to mention I did find an elementary proof, though it's a bit tedious. Its only remarkable feature is that in both it and my much shorter complex analytic proof, the final step is showing that x^2 - 3x + 3 > 0 for all real x. I'm not sure if there's a connection or if this is just a strange little coincidence. The two quadratics arise from very different manipulations so I lean towards coincidence. I haven't investigated further.