Mitochondrial DNA (mtDNA) and plastid DNA (ptDNA) encode vital bioenergetic apparatus, and mutations in these organelle DNA (oDNA) molecules can be devastating. In the germline of several animals, a genetic "bottleneck" increases cell-to-cell variance in mtDNA heteroplasmy, allowing purifying selection to act to maintain low proportions of mutant mtDNA. However, most eukaryotes do not sequester a germline early in development, and even the animal bottleneck remains poorly understood. How then do eukaryotic organelles avoid Muller's ratchet-the gradual buildup of deleterious oDNA mutations? Here, we construct a comprehensive and predictive genetic model, quantitatively describing how different mechanisms segregate and decrease oDNA damage across eukaryotes. We apply this comprehensive theory to characterise the animal bottleneck with recent single-cell observations in diverse mouse models. Further, we show that gene conversion is a particularly powerful mechanism to increase beneficial cell-to-cell variance without depleting oDNA copy number, explaining the benefit of observed oDNA recombination in diverse organisms which do not sequester animal-like germlines (for example, sponges, corals, fungi, and plants). Genomic, transcriptomic, and structural datasets across eukaryotes support this mechanism for generating beneficial variance without a germline bottleneck. This framework explains puzzling oDNA differences across taxa, suggesting how Muller's ratchet is avoided in different eukaryotes.