Accurate modeling of ridesourcing mode choices is essential for designing and implementing effective traffic management policies for reducing congestion, improving mobility, and allocating resources more efficiently. Existing models for predicting ridesourcing mode choices often suffer from limited predictive accuracy due to their inability to capture key psychological factors, and are further challenged by severe class imbalance, as ridesourcing trips comprise only a small fraction of individuals' daily travel. To address these limitations, this paper introduces the Synthesizing Attitudes, Predicting Actions (SAPA) framework, a hierarchical approach that uses Large Language Models (LLMs) to synthesize theory-grounded latent attitudes to predict ridesourcing choices. SAPA first uses an LLM to generate qualitative traveler personas from raw travel survey data and then trains a propensity-score model on demographic and behavioral features, enriched by those personas, to produce an individual-level score. Next, the LLM assigns quantitative scores to theory-driven latent variables (e.g., time and cost sensitivity), and a final classifier integrates the propensity score, latent-variable scores (with their interaction terms), and observable trip attributes to predict ridesourcing mode choice. Experiments on a large-scale, multi-year travel survey show that SAPA significantly outperforms state-of-the-art baselines, improving ridesourcing choice predictions by up to 75.9% in terms of PR-AUC on a held-out test set. This study provides a powerful tool for accurately predicting ridesourcing mode choices, and provides a methodology that is readily transferable to various applications.