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Electric vehicles are increasingly connected to distribution feeders whose transformers, cables, and voltage-control devices were not originally designed for coincident residential charging. Smart charging coordination can convert this challenge into an operational flexibility resource when user needs and network limits are treated jointly. This case study applies stochastic EV charging coordination to a realistic distribution-network testbed to reduce evening peaks, improve voltage profiles, and support local flexibility procurement. The objective is operational rather than purely algorithmic: the study evaluates what a distribution system operator and an aggregator can learn from coordinated charging under uncertain conditions. The case uses empirical-style charging-session statistics, a radial low-voltage feeder model, and stochastic scenarios for arrival time, departure time, energy demand, and feeder loading. Uncontrolled charging, deterministic coordination, and stochastic coordination are compared using the same network and mobility assumptions. The stochastic formulation is evaluated through network-level indicators for feeder peak demand, transformer loading, voltage deviation, flexibility availability, and procurement cost. The comparison is structured to show whether uncertainty-aware coordination is more robust than deterministic scheduling when charging behavior and grid conditions deviate from forecasts. Stochastic EV charging coordination is a practical planning and operational tool for distribution networks with rising residential and workplace charging demand. The case study provides implementation lessons for DSOs, aggregators, and flexibility-market designers.