LLM-based solvers have emerged as a promising means of automating problem modeling and solving. However, they remain unreliable and often depend on iterative repair loops that result in significant latency. We introduce OptiHive, an LLM-based framework that produces high-quality solvers for optimization problems from natural-language descriptions without iterative self-correction. OptiHive uses a single batched LLM query to generate diverse components (solvers, problem instances, and validation tests) and filters out erroneous components to ensure fully interpretable outputs. Taking into account the imperfection of the generated components, we employ a statistical model to infer their true performance, enabling principled uncertainty quantification and solver selection. On tasks ranging from traditional optimization problems to challenging variants of the Multi-Depot Vehicle Routing Problem, OptiHive significantly outperforms baselines, increasing the optimality rate from 5\% to 92\% on the most complex problems.