AI methods, such as generative models and reinforcement learning, have recently been applied to combinatorial optimization (CO) problems, especially NP-hard ones. This paper compares such GPU-based methods with classical CPU-based methods on the Maximum Independent Set (MIS) problem. Strikingly, even on in-distribution random graphs, leading AI-inspired methods are consistently outperformed by the state-of-the-art classical solver KaMIS running on a single CPU, and some AI-inspired methods frequently fail to surpass even the simplest degree-based greedy heuristic. Even with post-processing techniques like local search, AI-inspired methods still perform worse than CPU-based solvers. To better understand the source of these failures, we introduce a novel analysis, serialization, which reveals that non-backtracking AI-inspired methods, e.g. LTFT (which is based on GFlowNets), end up reasoning similarly to the simplest degree-based greedy, and thus worse than KaMIS. More generally, our findings suggest a need for a rethinking of current approaches in AI for CO, advocating for more rigorous benchmarking and the principled integration of classical heuristics. Additionally, we also find that CPU-based algorithm KaMIS have strong performance on sparse random graphs, which appears to show that the shattering threshold conjecture for large independent sets proposed by Coja-Oghlan & Efthymiou (2015) does not apply for real-life sizes (such as 10^6 nodes).