Randomization is a powerful and pervasive strategy for developing efficient and practical transmission scheduling algorithms in interference-limited wireless networks. Yet, despite the presence of a variety of earlier works on the design and analysis of particular randomized schedulers, there does not exist an extensive study of the limitations of randomization on the efficient scheduling in wireless networks. In this work, we aim to fill this gap by proposing a common modeling framework and three functional forms of randomized schedulers that utilize queue-length information to probabilistically schedule non-conflicting transmissions. This framework not only models many existing schedulers operating under a time-scale separation assumption as special cases, but it also contains a much wider class of potential schedulers that have not been analyzed. Our main results are the identification of necessary and sufficient conditions on the network topology and on the functional forms used in the randomization for throughput-optimality. Our analysis reveals an exponential and a sub-exponential class of functions that exhibit differences in the throughput-optimality. Also, we observe the significance of the network's scheduling diversity for throughput-optimality as measured by the number of maximal schedules each link belongs to. We further validate our theoretical results through numerical studies.