Some of the best thermoelectric (TE) materials to date are also topological insulators (TIs). While many studies have investigated the effects of topologically-protected TI surface states on TE properties, the conditions needed to realize such effects are quite different from typical operating conditions of TE devices for, e.g., power generation and room-temperature Peltier cooling. As a result, it is still unclear what properties of TIs, especially those related to the bulk band structure, are beneficial for TE performance, if any. Here, we perform high-throughput transport calculations using density functional theory to reveal that, within the same structure type, TIs tend to outperform normal insulators as TEs when properly optimized. The calculations also indicate that the TE performance is higher for TIs with strongly inverted bands. To explain these observations, we develop models based on Boltzmann transport theory which show that warping driven by band inversion, a key characteristic of TIs, is responsible for the high TE performance of TIs. We find that warping benefits the TE performance because of reduced transport mass and effectively higher valley degeneracy. Our results show that the band inversion strength is a critical property of a TI dictating the TE performance, and we suggest potential strategies to tune the inversion strength and enhance the TE performance in TIs, such as alloying and strain engineering. The study marks TIs as serious candidates for TE applications owing to band inversion-driven warping.