Self-centering energy-absorbing rocking core (SCENARIO) systems are emerging aseismic structural systems developed for obtaining excellent self-centering and collapse-resistant performance with negligible residual inter-story drifts in steel buildings under strong earthquakes. Despite its demonstrated superior structural performance, the nonstructural performance of SCENARIO systems, which is another essential consideration in seismic resilience evaluation of building structures, has not been well studied. For filling this research gap, this paper aims to make the original contributions by systematically assessing the structural and nonstructural damage of steel building structures equipped with SCENARIO systems under earthquakes and finding the efficient solution for enhancing the structural and nonstructural damage-control performance of SCENARIO systems. Two types of SCENARIO systems, including the SCENARIO system with friction spring dampers (denoted as SCENARIO-F system) and the SCENARIO system with hybrid dampers (denoted as SCENARIO-H system), were considered in this research. Three prototype steel buildings with SCENARIO systems were designed to achieve the same target displacement under design basis earthquakes (DBE). Incremental dynamic analyses (IDAs) were performed to study the seismic responses of the designed buildings subjected to far-field and near-fault ground motions with different intensities. The collapse-resistant performance, structural damage, post-earthquake repairability, and nonstructural damage of the designed buildings were investigated via seismic fragility analyses based on IDA results. Seismic fragility analysis results indicate that the SCENARIO-H systems show better performance than the SCENARIO-F systems in controlling structural and nonstructural damage under both far-field and near-fault earthquakes with high intensities. Benefiting from excellent self-centering capacity, the buildings equipped with SCENARIO-F and SCENARIO-H systems can show excellent post-earthquake repairability under strong earthquakes.