Stacked intelligent metasurfaces (SIMs) represent a key enabler for next-generation wireless networks, offering beamforming gains while significantly reducing radio-frequency chain requirements. In conventional space-only SIM architectures, the rate of reconfigurability of the SIM is equal to the inverse of the channel coherence time. This paper investigates a novel beamforming strategy for massive downlink connectivity using a randomized space-time (ST) coded SIM. In addition to conventional space-only metasurface layers, the proposed design integrates a ST metasurface layer at the input stage of the SIM that introduces random time variations over each channel coherence time interval. These artificial time variations enable opportunistic user scheduling and exploitation of multiuser diversity under slow channel dynamics. To mitigate the prohibitive overhead associated with full channel state information at the transmitter (CSIT), we propose a partial-CSIT-based beamforming scheme that leverages randomized steering vectors and limited user-side feedback based on signal quality measurements. Numerical results demonstrate that the proposed ST-SIM architecture achieves satisfactory sum-rate performance while significantly reducing CSIT acquisition and feedback overhead, thereby enabling scalable downlink connectivity in dense networks.