Two-dimensional materials can host topological phases stabilized either by an intrinsic moiré potential or topological crystallization. Crucial to the topology of these phases is the interplay between the (self-consistently generated) potential and the spinor structure, which imbues the band-projected potential with momentum-dependence. Truncating this potential at linear-order in momentum, we obtain a three-band model that captures the low-energy band structure and Berry curvature of such topological bands. We analytically find that the Berry curvature strongly depends on momentum-dependence of the potential. Furthermore, we identify that the model can be tuned to have emergent symmetries, including U(1) and anti-unitary symmetries. Emergent-symmetry breaking perturbations drive the system to regimes hosting topological flat bands, quadratic band touchings, and multiple Dirac cones. We recover many of these features when studying an un-truncated model of a λ-jellium spinor inside of a periodic potential. Our results reveal a rich interplay of spinor structure and a potential in determining the topology of the ground state.