The recent surge of interest in polaritons has prompted fundamental questions about the role of dark states in strong light-matter coupling phenomena. Here, we systematically vary the relative number of dark state polaritons by controlling the number of stacked CdSe nanoplatelets confined in a Fabry-Pérot cavity. We find the emission spectrum to change significantly with an increasing number of nanoplatelets, with a gradual shift of the dominant emission intensity from the lower polariton branch to a manifold of dark states. Through accompanying calculations based on a kinetic model, this shift is rationalized by an entropic trapping of excitations by the dark state manifold, while a weak dark state dispersion due to local disorder explains their non-zero emission. Our results point towards the relevance of the dark state concentration to the optical and dynamical properties of cavity-embedded quantum emitters with ramifications for Bose- Einstein condensate formation, polariton lasing, polariton-based quantum transduction schemes, and polariton chemistry.