We present a semiclassical analytic model for spherical core-shell surface plasmon lasers. Within this model, we drop the widely used one-mode approximations in favor of fully electromagnetic Mie theory. This allows for incorporation of realistic gain relaxation rates that so far are massively underestimated. Especially, higher order modes can undermine and even reverse the beneficial effects of the strong Purcell effect in such systems. Our model gives a clear view on gain and resonator requirements, as well as on the output characteristics that will help experimenters to design more efficient particle-based spasers.