We present a new approach for designing risk-bounded controllers for Uncrewed Aerial Vehicles (UAVs). Existing frameworks for assessing risk of UAV operations rely on knowing the conditional probability of an incident occurring given different causes. Limited data for computing these probabilities makes real-world implementation of these frameworks difficult. Furthermore, existing frameworks do not include control methods for risk mitigation. Our approach relies on UAV dynamics, and employs reachability analysis for a probabilistic risk assessment over all feasible UAV trajectories. We use this holistic risk assessment to formulate a control optimization problem that minimally changes a UAV's existing control law to be bounded by an accepted risk threshold. We call our approach PRReach. Public and readily available UAV dynamics models and open source spatial data for mapping hazard outcomes enables practical implementation of PRReach for both offline pre-flight and online in-flight risk assessment and mitigation. We evaluate PRReach through simulation experiments on real-world data. Results show that PRReach controllers reduce risk by up to 24% offline, and up to 53% online from classical controllers.