Kinetic proofreading is an error-correcting mechanism by which biology expends energy to avoid mistakes during processes like replication and translation. Proofreading is typically assumed to evolve due to a selection for replication fidelity, outweighing presumed costs in speed and energy. However, incorporating mismatches often stalls replication. We show that when stalling is accounted for, proofreading can instead speed up replication. We generalize our findings to multi-component self-assembly, and show that analogous error-correcting processes, such as a dynamic instability, can similarly emerge solely due to selection for rapid assembly. Our work shows that non-equilibrium error-correction can evolve solely from a selection for speed, independent of advantages associated with fidelity. We explore the possibility of the evolution of error correction through speed in a DNA polymerase homologous to phi29. We repurpose a directed evolution platform in yeast, OrthoRep, to characterize speed and error rate of a DNA polymerase. We develop high-throughput assays to measure error rate and speed of all single mutants of the polymerase gene with a double barcoded library. This platform allows access to the two relevant traits, speed and error rate, in high throughput and provides the largest mutagenesis screen of a polymerase and for which both phenotypes are measured. This enables to corroborate our hypothesis of evolution of proofreading through speed and identify the control knobs for engineering polymerases selecting for either of the two traits, speed and error rate.