Multi-agent LLM pipelines produce contradictory evidence on whether team diversity improves output quality: heterogeneous Mixture-of-Agents teams outperform single models, yet homogeneous Self-MoA teams consistently win under synthesis-based aggregation. We propose a resolution by identifying the selection bottleneck—a crossover threshold in aggregation quality that determines whether diversity helps or hurts. Under this model, we obtain a closed-form crossover threshold s∗ (Proposition 1) that separates the regimes where diversity helps and hurts. In a targeted experiment spanning 42 tasks across seven categories (N=210), a diverse team with judge-based selection achieves a win rate of 0.810 against a single-model baseline, while a homogeneous team scores 0.512—near chance (Glass’s Δ=2.07). Judge-based selection outperforms MoA-style synthesis by ΔWR=+0.631—the synthesis approach is preferred over the baseline in zero of 42 tasks by the judge panel. A decoupled evaluation with fully independent judges confirms all directional findings (Spearman ρ=0.90); win rates attenuate by 53–67% under independent evaluation relative to the primary estimates, consistent with partial measurement circularity in the judge-based cells. Exploratory evidence suggests that including a weaker model improves performance while reducing cost (p<10−4, not pre-registered). Our results suggest that selector quality may be a more impactful design lever than generator diversity in single-round generate-then-select pipelines, with a specific Opus-only singleton as the baseline.