Min-max routing problems aim to minimize the maximum tour length among agents as they collaboratively visit all cities, i.e., the completion time. These problems include impactful real-world applications but are known as NP-hard. Existing methods are facing challenges, particularly in large-scale problems that require the coordination of numerous agents to cover thousands of cities. This paper proposes a new deep-learning framework to solve large-scale min-max routing problems. We model the simultaneous decision-making of multiple agents as a sequential generation process, allowing the utilization of scalable deep-learning models for sequential decision-making. In the sequentially approximated problem, we propose a scalable contextual Transformer model, Equity-Transformer, which generates sequential actions considering an equitable workload among other agents. The effectiveness of Equity-Transformer is demonstrated through its superior performance in two representative min-max routing tasks: the min-max multiple traveling salesman problem (min-max mTSP) and the min-max multiple pick-up and delivery problem (min-max mPDP). Notably, our method achieves significant reductions of runtime, approximately 335 times, and cost values of about 53% compared to a competitive heuristic (LKH3) in the case of 100 vehicles with 1,000 cities of mTSP.