Terminal platelet production and platelet activation involve considerable cytoskeletal reorganization. The morphological changes brought about by this remodeling are essential for proper platelet function. Recent discoveries suggest that the marginal band of microtubules found at the equator of platelets and certain platelet precursors, undergoes a similar instability. This buckling instability serves, in one case, as the final differentiator of platelet production, and in the other, as an early step in the activation pathway. We aim to understand the mechanical principles governing these processes. To accomplish this, we construct a theoretical framework describing a growing, elastic ring confined within a flexible vesicle. With this method we construct two phase diagrams. The first corresponds to an instability due the curvature elasticity of the membrane, and the second to the same instability, but resulting from the strain energy of the membrane cortex. Next, we develop a coarse-grained Monte Carlo model of a growing marginal band within the platelet/preplatelet cytoskeleton. With this model we observe that the confining membrane suppresses the instability more readily for smaller marginal bands, thereby increasing the stability of platelets compared to their precursors.