Adaptive evolution depends on the supply of heritable variation, yet excessive mutation threatens viability by degrading essential molecular functions. Here, we show that this trade-off emerges naturally from the kinetic proofreading mechanism that controls replication fidelity. In our model, environmental shifts alter the optimal driving rate constant of proofreading enzymes, transiently elevating replication error rates and triggering rapid evolutionary change until a new fidelity optimum is reached. This produces alternating periods of stasis and rapid adaptation, consistent with punctuated equilibrium. We further show that coding-region length and population size jointly determine whether adaptation succeeds or mutational collapse occurs, reflecting the balance between mutation supply and error tolerance predicted by the drift-barrier principle. These results provide a molecular mechanistic basis for nearly neutral evolutionary dynamics and illustrate how genomic organization constrains long-term evolutionary resilience.