Agentic pipelines for structured-query generation are rapidly expanding, but it is unclear which part of the loop produces the gain. We use LAST-CQ -- a five-agent, training-free, execution-grounded Text-to-Cypher framework -- as an instrumented testbed, running three counterfactuals over 2,471 live-database queries and six backbones spanning three vendor scale tiers. Removing correction is worth between 3.1% aggregate execution-BLEU against the single-pass system and 12.3% against a no-refinement counterfactual (up to 80.7% for the weakest backbone). Replacing schema-grounded, LLM-synthesised feedback with raw database error strings costs almost nothing (20.9% vs. 19.9% naive exact match; <0.2% end-to-end; equivalent within
±0.075 set-F1 by two one-sided tests). Spending the same call budget on parallel sampling degrades quality by 10-11%. What works is detecting failure and routing it to a retry, not the feedback sophistication or number of samples. LAST-CQ itself recovers 91.7% of queries that fail under single-pass generation, while a query that succeeds first time still costs exactly one LLM call. We also show that n-gram overlap on serialised results is not a bound in either direction: it over-scores against set equivalence on 65.9% of results while under-scoring against judged semantics. Finally, we calibrate our LLM judge against blind human labels and find it optimistic by 9 points.