In clinical decision-making, predictive models face a persistent trade-off: accurate models are often opaque "black boxes," while interpretable methods frequently lack predictive precision or statistical grounding. In this paper, we challenge this dichotomy, positing that high predictive accuracy and high-quality descriptive explanations are not competing goals but synergistic outcomes of a deep, first-hand understanding of data. We propose the Reflective Cognitive Architecture (RCA), a novel framework designed to enable Large Language Models (LLMs) to learn directly from tabular data through experience and reflection. RCA integrates two core mechanisms: an iterative rules optimization process that refines logical argumentation by learning from prediction errors, and a distribution-aware rules check that grounds this logic in global statistical evidence to ensure robustness. We evaluated RCA against over 20 baselines - ranging from traditional machine learning to advanced reasoning LLMs and agents - across diverse medical datasets, including a proprietary real-world Catheter-Related Thrombosis (CRT) cohort. Crucially, to demonstrate real-world scalability, we extended our evaluation to two large-scale datasets. The results confirm that RCA achieves state-of-the-art predictive performance and superior robustness to data noise while simultaneously generating clear, logical, and evidence-based explanatory statements, maintaining its efficacy even at scale. The code is available at https://github.com/ssssszj/RCA.