Graph Transformer (GT) has recently emerged as a promising neural network architecture for learning graph-structured data. However, its global attention mechanism with quadratic complexity concerning the graph scale prevents wider application to large graphs. While current methods attempt to enhance GT scalability by altering model architecture or encoding hierarchical graph data, our analysis reveals that these models still suffer from the computational bottleneck related to graph-scale operations. In this work, we target the GT scalability issue and propose DHIL-GT, a scalable Graph Transformer that simplifies network learning by fully decoupling the graph computation to a separate stage in advance. DHIL-GT effectively retrieves hierarchical information by exploiting the graph labeling technique, as we show that the graph label hierarchy is more informative than plain adjacency by offering global connections while promoting locality, and is particularly suitable for handling complex graph patterns such as heterophily. We further design subgraph sampling and positional encoding schemes for precomputing model input on top of graph labels in an end-to-end manner. The training stage thus favorably removes graph-related computations, leading to ideal mini-batch capability and GPU utilization. Notably, the precomputation and training processes of DHIL-GT achieve complexities linear to the number of graph edges and nodes, respectively. Extensive experiments demonstrate that DHIL-GT is efficient in terms of computational boost and mini-batch capability over existing scalable Graph Transformer designs on large-scale benchmarks, while achieving top-tier effectiveness on both homophilous and heterophilous graphs.