Preventing progressive collapse induced by accidental events poses a critical challenge in the design and construction of resilient structures. While substantial progress has been made in planar structures, the progressive collapse mechanisms of precast concrete spatial structures—particularly regarding the effects of precast slabs—remain inadequately explored. This study develops a refined finite element modeling approach to investigate progressive collapse mechanisms in fully bonded prestressed precast concrete (FB-PPC) spatial frames, both with and without precast slabs. The modeling approach was validated against available test data from related sub-assemblies, and applied to assess the collapse performance. A series of pushdown analyses were conducted on the spatial frames under various column removal scenarios. The load–displacement curves, slab contribution, and failure modes under different conditions were compared and analyzed. A simplified energy-based dynamic assessment was additionally employed to offer a rapid estimation of the dynamic collapse capacity. The results show that when interior or side columns fail, the progressive collapse process can be divided into the beam action stage and the catenary action (CA) stage. During the beam action stage, the compressive membrane action (CMA) of the slabs and the compressive arch action (CAA) of the beams work in coordination. Additionally, the tensile membrane action (TMA) of the slabs strengthens the CA in the beams. When the corner columns fail, the collapse stages comprise the beam action stage followed by the collapse stage. Due to insufficient lateral restraints around the failed column, the development of CA is limited. The membrane action of the slabs cannot be fully mobilized. The contribution of the slabs is significant, as it can substantially enhance the vertical resistance and restrain the lateral displacement of the columns. The energy-based dynamic assessment further reveals that FB-PPC spatial frames exhibit high ductility and residual strength following sudden column removal, with dynamic load–displacement curves showing sustained plateaus or gentle slopes across all scenarios. The inclusion of precast slabs consistently enhances both the peak load capacity and the residual resistance in dynamic collapse curves.