Large Language Models (LLMs) have revolutionized Natural Language to SQL (NL-to-SQL), dominating most NL-to-SQL benchmarks. But LLMs still face limitations due to hallucinations, semantic ambiguity, and lexical mismatches between an NL query and the database schema. Naturally, a lot of work in the ML+DB intersection aims to mitigate such LLM limitations. In this work, we shine the light on a complementary data-centric question: How should DB schemas evolve in this era of LLMs to boost NL-to-SQL? The intuition is that more NL-friendly schema identifiers can help LLMs work better with DBs. We dive deeper into this seemingly obvious, but hitherto underexplored and important, connection between schema identifier ''naturalness'' and the behavior of LLM-based NL-to-SQL by creating a new integrated benchmark suite we call SNAILS. SNAILS has 4 novel artifacts: (1) A collection of real-world DB schemas not present in prior NL-to-SQL benchmarks; (2) A set of labeled NL-SQL query pairs on our collection not seen before by public LLMs; (3) A notion of naturalness level for schema identifiers and a novel labeled dataset of modified identifiers; and (4) AI artifacts to automatically modify identifier naturalness. Using SNAILS, we perform a comprehensive empirical evaluation of the impact of schema naturalness on LLM-based NL-to-SQL accuracy, and present a method for improving LLM-based NL-to-SQL with natural views. Our results reveal statistically significant correlations across multiple public LLMs from OpenAI, Meta, and Google on multiple databases using both zero-shot prompting as well as more complex NL-to-SQL workflows: DIN SQL, and CodeS. We present several fine-grained insights and discuss pathways for DB practitioners to better exploit LLMs for NL-to-SQL.