Persistent currents in fermionic superfluids exhibit fundamentally different stability behavior than their bosonic counterparts. Using time-dependent density functional theory for ultracold Fermi gases in the BCS regime, we demonstrate that superflow stability is intrinsically limited by a pair-breaking threshold - a critical velocity where Cooper pairs dissociate. This mechanism is analogous to the depairing current in superconductors, which sets the intrinsic upper bound on supercurrent density. Above this threshold, impurities cannot stabilize the flow, regardless of their number or arrangement. This contrasts sharply with Bose-Einstein condensates, where the addition of impurities can enhance current stability by suppressing phase slips. We identify two dissipation channels: topological (vortex emission) and microscopic (pair-breaking). Crucially, pair-breaking drives dissipation even when no vortices are present and topology remains stable. Impurity size and spacing control vortex mobility, revealing regimes of collective pinning and vortex hopping between sites. These findings extend beyond ultracold atoms to all weak-coupling fermionic superfluids. The pair-breaking mechanism and impurity-mediated vortex dynamics are applicable to both superconductors and neutron star crusts, providing insight into critical current limitations and pulsar glitches.