Unsupervised plain graph alignment (UPGA) aims to align corresponding nodes across two graphs without any auxiliary information. Existing UPGA methods rely on structural consistency while neglecting the inherent structural differences in real-world graphs, leading to biased node representations. Moreover, their one-shot alignment strategies lack mechanisms to correct erroneous matches arising from inaccurate anchor seeds. To address these issues, this paper proposes IterAlign, a novel parameter-free and efficient UPGA method. First, a simple yet powerful representation generation method based on heat diffusion is introduced to capture multi-level structural characteristics, mitigating the over-reliance on structural consistency and generating stable node representations. Two complementary node alignment strategies are then adopted to balance alignment accuracy and efficiency across graphs of varying scales. By alternating between representation generation and node alignment, IterAlign iteratively rectifies biases in nodes representations and refines the alignment process, leading to superior and robust alignment performance. Extensive experiments on three public benchmarks demonstrate that the proposed IterAlign outperforms state-of-the-art UPGA approaches with a lower computational overhead, but also showcases the ability to approach the theoretical accuracy upper bound of unsupervised plain graph alignment task.