Automated Program Repair (APR) agents leverage Large Language Models (LLMs) to autonomously diagnose and fix software bugs through reasoning, planning, and tool use. Despite impressive leaderboard gains on benchmarks such as SWE-bench, little is understood about how these agents take actions, where they fail, and how their behavior compares to that of human developers. This paper presents the first systematic analysis of five state-of-the-art APR agents across 500 real-world repair tasks, tracing their full decision-making pipelines -- from issue description to patch validation. Our study reveals that while agents excel at simple fixes, they struggle with logic-intensive bugs, often producing verbose or overfitted patches that merely satisfy existing tests. We find that test generation and regression test selection remain major bottlenecks, with agents frequently failing to reproduce issues or run relevant regression tests. Moreover, most agents operate with primitive tooling (e.g., bash scripts) and lack access to debuggers or program analyzers, which constrains their reasoning and patch quality. These findings highlight key limitations in current APR systems and motivate a shift-left approach -- emphasizing early, high-quality test generation and validation -- to reduce spurious fixes and improve semantic correctness. We further outline concrete directions for next-generation APR design: (1) richer and more integrated tool ecosystems, (2) diversified agentic architectures that combine complementary strengths, and (3) benchmarks that prioritize semantic repair quality and test generation fidelity over surface-level success metrics.