Sparse matrix-vector multiplication (SpMV) is a fundamental operation in scientific computing, machine learning, and graph analytics, demanding efficient execution on modern hardware. Recent advances in hardware accelerators, such as Tensor Cores, have significantly improved the performance of many compute-intensive workloads. However, effectively utilizing Tensor Cores for SpMV remains challenging due to its irregular sparsity patterns and the mismatch between SpMV's computational characteristics and constrained architecture design, leading to suboptimal performance and underutilization of Tensor Cores. In this paper, we systematically analyze the state-of-the-art SpMV optimizations on Tensor Cores, identify key performance bottlenecks, and propose Drawloom, a Tensor-Core-aware framework for SpMV with efficient Tensor Core mapping and optimized pipeline execution. Drawloom leverages a redesigned Tensor Core mapping strategy with a zig-zag chained sparse storage format, as well as a multi-stage register pipeline to better exploit hardware parallelism. Our evaluation on SuiteSparse dataset demonstrates that Drawloom outperforms cuSPARSE by 2.71$\times$/1.90$\times$ (in FP16), {2.95$\times$/2.39$\times$ (in FP32)}, and 2.47$\times$/1.54$\times$ (in FP64) on A100 and H100 GPUs, respectively. Compared to the state-of-the-art SpMV implementations, Drawloom achieves a performance speedup of 1.26$\times$/1.18$\times$ (in FP16) and 1.49$\times$/1.56$\times$ (in FP64) on A100 and H100 GPUs, respectively.