Graph convolutional networks (GCNs) are prevalent in graph learning. Hybrid architecture, comprised of interconnected engines, is widely used in accelerator designs for GCNs to handle their hybrid execution patterns. However, its inherent inefficiency arises from local bottlenecks in each engine, driven by fluctuating compute resource requirements, thereby causing low overall compute utilization. In this work, we propose a novel GCN accelerator with unified architecture called uFlowGCN, which is in-cooperated with reduction-tree-based computing unit design and tag-driven scheduling. uFlowGCN enables the efficient parallel execution of hybrid execution patterns on a unified hardware substrate, allowing for dynamic allocation of compute resources on the fly to alleviate local bottlenecks. Compared to the optimal design of hybrid architecture, uFlowGCN achieves up to 4.3 \(\times \) speedup and 95.3% \(\sim \) 99.5% compute utilization.