Bismuth has recently attracted interest in connection with Na-ion battery anodes due to its high volumetric capacity. It reacts with Na to form Na ${}_3$ Bi which is a prototypical Dirac semimetal with a nontrivial electronic structure. Density-functional-theory based first-principles calculations are playing a key role in understanding the fascinating electronic structure of Na ${}_3$ Bi and other topological materials. In particular, the strongly-constrained-and-appropriately-normed (SCAN) meta-generalized-gradient-approximation (meta-GGA) has shown significant improvement over the widely used generalized-gradient-approximation (GGA) scheme in capturing energetic, structural, and electronic properties of many classes of materials. Here, we discuss the electronic structure of Na ${}_3$ Bi within the SCAN framework and show that the resulting Fermi velocities and s-band shift around the $\Gamma$ point are in better agreement with experiments than the corresponding GGA predictions. SCAN yields a purely spin-orbit-coupling (SOC) driven Dirac semimetal state in Na ${}_3$ Bi in contrast with the earlier GGA results. Our analysis reveals the presence of a topological phase transition from the Dirac semimetal to a trivial band insulator phase in Na ${}_3$ Bi ${}_x$ Sb ${}_{1-x}$ alloys as the strength of the SOC varies with Sb content, and gives insight into the role of the SOC in modulating conduction properties of Na ${}_3$ Bi.