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This paper presents an end-to-end framework for safe learning-based control (LbC) using nonlinear stochastic MPC and distributionally robust optimization (DRO). This work is motivated by several open challenges in LbC literature. In particular, many control-theoretic LbC methods require subject matter expertise in order to translate their own safety guarantees, often manifested as preexisting data of safe trajectories or structural model knowledge. In this paper, we focus on LbC where the controller is applied directly to a system of which it has no or extremely limited direct experience, towards safety during \textit{tabula-rasa} or ``\textit{blank slate''} model-based learning and control as a challenging case for validation. This explores the boundary of the status-quo in control theory relating to requirements for subject matter expertise. We show under basic and limited assumptions on the underlying problem, we can translate probabilistic guarantees on feasibility to nonlinear systems using results in stochastic MPC and DRO literature whose relevance we formally extend in a mathematical analysis. We also present a coupled and intuitive formulation for persistence of excitation (PoE), and illustrate the connection between PoE and applicability of the proposed method. Our case studies of vehicle obstacle avoidance and safe extreme fast charging of lithium-ion batteries reveal powerful empirical results supporting the underlying DRO theory. Our method is widely applicable within the LbC domain to, for example, airborne wind energy systems, vehicle obstacle avoidance, and energy storage systems management. It is also applicable to quantifying uncertainty beyond the LbC case.