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Transformers have been widely regarded as a promising direction for breaking through the performance bottlenecks of Graph Neural Networks (GNNs), primarily due to their global receptive fields. However, a recent empirical study suggests that tuned classical GNNs can match or even outperform state-of-the-art Graph Transformers (GTs) on standard node classification benchmarks. Motivated by this fact, we deconstruct several representative GTs to examine how global attention components influence node representations. We find that the global attention module does not provide significant performance gains and may even exacerbate test error oscillations. Consequently, we consider that the Transformer is barely able to learn connectivity patterns that meaningfully complement the original graph topology. Interestingly, we further observe that mitigating such oscillations enables the Transformer to improve generalization in GNNs. In a nutshell, we reinterpret the Transformer through the lens of graph spectrum and reformulate it as a global-aware graph filter with band-pass characteristics and linear complexity. This unique perspective introduces multi-channel filtering constraints that effectively suppress test error oscillations. Extensive experiments (17 homophilous, heterophilous graphs) provide comprehensive empirical evidence for our perspective. This work clarifies the role of Transformers in GNNs and suggests that advancing modern GNN research may still require a return to the graph itself.