Graph Neural Networks (GNNs) have become essential in high-stakes domains such as drug discovery, yet their black-box nature remains a significant barrier to trustworthiness. While self-explainable GNNs attempt to bridge this gap, they often rely on standard message-passing backbones that inherit fundamental limitations, including the 1-Weisfeiler-Lehman (1-WL) expressivity barrier and a lack of fine-grained interpretability. To address these challenges, we propose SymGraph, a symbolic framework designed to transcend these constraints. By replacing continuous message passing with discrete structural hashing and topological role-based aggregation, our architecture theoretically surpasses the 1-WL barrier, achieving superior expressiveness without the overhead of differentiable optimization. Extensive empirical evaluations demonstrate that SymGraph achieves state-of-the-art performance, outperforming existing self-explainable GNNs. Notably, SymGraph delivers 10x to 100x speedups in training time using only CPU execution. Furthermore, SymGraph generates rules with superior semantic granularity compared to existing rule-based methods, offering great potential for scientific discovery and explainable AI.